The Main Theorems of Calculus
It is better to state the completeness property which is the topic of the question.
Completeness property of the real number system is the property of real numbers which distinguishes it from the rational numbers.
Apart from this property both the real numbers and rational numbers behave in exactly the same manner. The property can be expressed in many forms (and I am not sure if you can understand all the forms):
Dedekind's Theorem: If all the real numbers are grouped into two non-empty sets $L$ and $U$ such that $L \cup U = \mathbb{R}, L \cap U = \emptyset$ and further if every member of $L$ is less than every member of $U$, then there is a unique real number $\alpha$ such that all real numbers less than $\alpha$ belong to $L$ and all real numbers greater than $\alpha$ belong to $U$.
Least upper bound property: If $A$ is a non-empty set of real numbers such that no member of $A$ exceeds a constant real number $K$ (say), then there is a real number $M$ with the property that no member of $A$ exceeds $M$, but every real number less than $M$ is exceeded by at least one member of $A$. This number $M$ is called the least upper bound or supremum of $A$.
Cauchy Criterion of Completeness: Let $\{a_{n}\} $ be a sequence of real numbers such that for any given real number $\epsilon > 0$ there is a positive integer $n_{0}$ such that $|a_{n} - a_{m}| < \epsilon$ for all positive integers $m, n$ with $m \geq n_{0}, n \geq n_{0}$. Then the sequence $a_{n}$ converges to some real number $L$. In other words, there exists a number $L$ such that for any given $\epsilon > 0$ there is a positive integer $n_{0}$ such that $|a_{n} - L| < \epsilon$ for all $n \geq n_{0}$.
- Nested Interval Principle: Let $I_{n} = [a_{n}, b_{n}]$ be a sequence of closed intervals such that $I_{n + 1} \subseteq I_{n}$ then there is at least one real number $\alpha$ which lies in all the intervals $I_{n}$ i.e. $\bigcap_{n = 1}^{\infty}I_{n} \neq \emptyset$. This is sometimes also called Cantor's Intersection Theorem.
- Bolzano-Weierstrass Theorem: If $S$ is an infinite set of real numbers which is bounded then there is a real number $c$ (which may or may not belong to $S$) such that every neighbourhood of $c$ contains a point of $S$ other than $c$. Such a point $c$ is called a limit point or an accumulation point of $S$.
- Heine Borel Principle: Let $[a, b]$ be a closed interval and let $\mathcal{C}$ be a collection of open intervals $I$ such that each point of $[a, b]$ lies in some interval $I \in \mathcal{C}$. Then it is possible to choose a finite number of intervals from $\mathcal{C}$ say $I_{1}, I_{2}, \ldots, I_{n}$ such that each point of $[a, b]$ lies in some interval $I_{j}$.
- Absolute Convergence Principle: If $\{a_n\} $ is a sequence then the infinite series $\sum a_n$ is said to be convergent with sum $s$ if the sequence $\{s_n\} $ defined by $$s_n=a_1+a_2+\dots+a_n$$ converges to $s$. If the series $\sum |a_n|$ is convergent then the series $\sum a_n$ is said to be absolutely convergent. Absolute convergence of an infinite series implies its convergence. (This version of completeness was discussed by user @Shahab in comments and has a non-trivial and non-obvious proof.)
None of the above results hold if the numbers concerned are changed into rational numbers. Moreover assuming any one of the above properties, the others can be proved. These are all different notions of completeness which are equivalent in the real number system. Also note that all these properties are sort of existence theorems in the sense that they guarantee the existence of something useful in certain particular contexts.
Out of these the simplest one to understand for a calculus beginner (any student of age 15-16 years) is the first property called Dedekind's Theorem named after Richard Dedekind who developed a rigorous theory of real numbers.
What Taylor's book is saying is very correct and perhaps very very unfortunate for the students who are learning calculus. The subject of calculus builds up on the foundations of real number system and it is a shame that before learning calculus the only machinery available to students is the concept of rational numbers, laws of algebra to manipulate expressions and some knowledge about specific types of irrational numbers like surds or $\pi$.
Any significant theorem of calculus (i.e. theorems which don't deal with algebraic manipulation of various things in calculus) is proven using the completeness property of real numbers. OP wants to know: why is it so?
Well it is difficult to answer your question directly. In my opinion one should say that any theorem of calculus which is based on the completeness property is significant and any theorem of calculus which is not based on the completeness property is insignificant in the sense that it can be obtained by the use of properties of rational numbers and laws of algebra and thereby these are nothing but extensions of algebra in a particular direction.
Thus a theorem like $$\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$$ is insignificant and is more of an extension of algebra. But the theorem which says that a sequence which is increasing and bounded above converges to a limit is significant. Rolle's Theorem, Mean Value Theorem, Intermediate Value Theorem etc are all significant theorems. It is precisely because the "completeness property" is never introduced in many calculus textbooks that these theorems of calculus are never proved in such textbooks.
For integral calculus too, the story is much worse similar. Guaranteeing the existence of anti-derivative for continuous functions is a difficult result which can't be proved without the completeness property (it uses the Heine Borel Principle in one of the proofs). However you can pretty much invert the differentiation formulas for elementary functions to get anti-derivatives for common functions and this is fine without using completeness.
Finally there is the famous Fundamental Theorem of Algebra (namely that any polynomial of positive degree with complex coefficients has a complex root) which can't be proved without completeness property. This shows that the theorem actually belongs to analysis and is named so because of historical reasons. It does not serve so much of a fundamental purpose in algebra.
Although you did not ask but it is important to throw some light on "why the completeness property is the missing ingredient in most calculus courses". It is because before learning calculus the students are mostly trained in algebraic manipulation of mathematical expressions (along with some basic theorems in Euclidean Geometry) and various textbook authors and instructors think that it will be difficult for students to depart from the algebraic manipulation involving $+,-,\times, /$ and focus on order relations involving $<, >$ (which is true to some extent for many of the students). The completeness property can not even be understood properly (leave its proof) if one does not appreciate the concept of order relations.
At the same time the same textbook authors and instructors feel that it is best if calculus can also be presented as algebraic manipulation of somewhat complicated and new things then the students would sail through a calculus course easily without the need to focus on order relations and completeness property and one champion of this thought is Walter Rudin. On the other hand there are many textbooks like "Principles of Mathematical Analyis" by Rudin which just assume such an important property as completeness as an axiom and justify this approach on the basis of "sound pedagogy".
Such an approach keeps the students limited to their old techniques and attitudes of algebra and they never understand that the fundamental thing to learn in calculus is to appreciate the concept of order relations and thereby study the proper theory of real numbers and the concepts like limit, continuity, derivative and integral based on the theory of real numbers. With this approach calculus seems very mysterious / magical and yes very confusing to students who are new to it. Plus they miss the understanding of all the major and fundamental theorems of calculus.
I now quote my favorite example. In order to learn calculus properly/seriously a student must be able to prove the following two statements:
- There is no rational number whose square is $2$.
- If $a$ is a positive rational number such $a^{2} < 2$ then there is a rational number $b > a$ such that $b^{2} < 2$.
The first result belongs to algebra and it is a routine exercise in high school. The second result is the beginning of real-analysis (without mentioning or using anything about real numbers) and if the student can sail through this result, then he can very well learn a proper theory of real numbers via Dedekind's cuts and for him calculus is never going to be mysterious / confusing.
Fortunately such an approach which develops theory of real numbers before developing calculus is available in G. H. Hardy's classic textbook "A Course of Pure Mathematics."
Suppose we tried to do calculus using only the rational numbers. Define $f$ so that $f(x) = 0$ if $x^2 < 2$, and $f(x) = 1$ otherwise. Then $f'(x) = 0$ for all $x$, but $f$ is not constant. This is the kind of thing that goes wrong without the completeness axiom.
Completeness – whether defined in terms of Cauchy sequences, or in terms of existence of $\inf$ and $\sup$ – is a property of the fine structure of ${\mathbb R}$. It allows you to prove the existence of interesting limits without knowing these limits beforehand. Some examples:
We don't need completeness to prove that $$\lim_{x\to1}{x^2-1\over x-1}=2,\qquad{\rm or}\qquad\sum_{k=0}^\infty 2^{-k}=2\ ,$$ because we guessed all along that the limit in these cases is $2$, and it is easy to do some $\epsilon$-$\delta$-chasing in order to provide a proof. But it is another thing with the limit $$\lim_{n\to\infty}\left(1+{1\over n}\right)^n\ .\tag{1}$$ Here completeness does the following for us: If the sequence $a_n:=\left(1+{1\over n}\right)^n$ satisfies certain criteria (monotonicity, boundedness), which can be checked in finite time, i.e., by looking at the given $a_n$ alone, then the limit $(1)$ exists. Similarly for the limit of the sum $$R_n:=\sum_{k=1}^n{1\over n+k}\ ,$$ which is a Riemann sum for the integral $\int_1^2{1\over x}\>dx=\log 2$.