Doubt on rational and real numbers

You could think about it this way: there's no real number $x$ such that $x^2 = -1$, but that doesn't imply $\mathbb{R}$ has holes. It just means that there are things beyond $\mathbb{R}$. The point of the second assertion is to show that not only $\mathbb{Q}$ isn't complete (in an everyday sense, in the same way that $\mathbb{R}$ isn't complete because you can extend it to $\mathbb{C}$), but in some sense you can split the whole of it into two parts and leave a gap in between.

What the text may be trying to assert is that the rational numbers can be split into two "separate" pieces. The second claim shows two such pieces, and notes that no point of either piece is "infinitely close" to the other piece (e.g. we would say that $0$ is infinitely close to the set of positive numbers, since we can find positive numbers arbitrarily close to $0$). To show that their union is all of $\mathbb{Q}$, we need the first claim, that the element we have "left out" (that is $\sqrt{2}$), is not in $\mathbb{Q}$.

Does the textbook talk about the Intermediate Value Theorem anywhere nearby? It may be trying to show that it does not hold in $\mathbb{Q}$.

Let $f(x) = x^2$. It is continuous, even in $\mathbb{Q}$. Since $2$ is between $f(1)$ and $f(2)$, it would seem intuitive that there exists a $c \in [1, 2]$ such that $f(c) = 2$. However, unlike in $\mathbb{R}$, this is not true.