How to appreciate Fermat's last theorem?

Clearly there are real and algebraic solutions. Just pick any value for $x$ and $y$ you want, and then solve for $z$ (possibly in the complex numbers). Since integers and rationals were the "first numbers" and Diophantine equations perhaps the "first equations" it makes some sense that we search for rational and integer solutions to Diophantine equations. In this case, because of homogeneity (each term in the equation has the same degree), rational solutions imply integer solutions (just clear denominators to get integer solutions from rational ones as so:

$$\left(\frac{a}{b}\right)^n+\left(\frac{c}{d}\right)^n=\left(\frac{e}{f}\right)^n \iff (adf)^n+(cbf)^n=(ebd)^n.$$

And the existence of integer solutions implies the existence of whole number solutions. First, if the exponent $n$ is even, then any solution $(x,y,z)$ means there is a solution $(|x|,|y|,|z|)$. Second, if $n$ is odd and for instance $(-x)^n+y^n=z^n$ then $y^n=x^n+z^n$ hence $(x,z,y)$ is a whole number solution, or else if $(-x)^n+(-y)^n=x^n$ then $x^n+y^n+z^n=0$ which isn't possible for nonzero $(x,y,z)$ so that case can't occur anyway. Any integer solution $(x,y,z)$ will yield a whole number solution $(a,b,c)$ by negating or permuting the $x,y,z$ as appropriate (try it out!).

Among Diophantine equations, $x^n+y^n=z^n$ is at once both very simple in form and surprisingly deep in its truth (if the length and complexity of its proof are any indication), which sets it apart from other Diophantine equations. Personally, I'm not terribly interested in all of these Diophantine equations, and in fact I find the Modularity theorem (which was a critical component in the proof of Fermat's last theorem, and previously called the Taniyama-Shimura-Weil conjecture) more interesting. But this illustrates an important point that is touched on again and again: the search for a proof of Fermat's last theorem is what inspired the development of a lot of modern algebraic number theory, and so the theorem is very significant in terms of inspiration and history.

Integer solutions of a problem are essential in many areas. You wouldn't like that the number of persons to fit in the room be $13.5$, would you ?

But besides this obvious pragmatism, it turns out that all research around integer solutions of equations, the so-called Number theory, ends-up in numerous beautiful (and difficult) extensions that make the delight of mathematicians.

Fermat's last theorem is a perfect example, having required an army of the finest minds for several centuries before the answer was found.

Interestingly, Number theory makes a heavy use of techniques drawn from the theory of real and complex variables.

Do you know the website Wolfram Alpha? You can use it to play around with equations, which will perhaps deepen your appreciation of Fermat's so-called last theorem (both words are problematic, but that's a different can of worms).

Part of the reason whole numbers are important is because the problem becomes trivially simple if the requirement for integers is dropped. Consider for example $n = 3$. Go to Wolfram Alpha and type in solve x^3 + (5/2)^3 = 7^3. One of the three solutions is $$\frac{3 \root 3 \of {97}}{2}.$$ Now change (5/2) to (46/7). Play around with it some more. You'll become convinced that, without the requirement of integers, there's always a non-trivial solution.

Then try this: solve x^3 + y^3 = 7^3 in integers. There are solutions, but these require $x$ or $y$ to be zero, which is no fun, right?