[linear Algebra] Can someone please explain what the textbook answer for proving non-inverse means?

>I also know that the question to check for invertibility means to check if the function is 1-to-1 and onto.

That is not what the question is asking. They are asking you whether d/dx and ∫ are inverse **to each other**, not just whether they are invertible at all. To say that two functions f,g are inverse to each other means that f∘g and g∘f are both the identity map on their respective domains. The answer is showing that ∫(d/dx(1))≠1, ie, ∫∘d/dx is not the identity map.
Two functions f(x) and g(x) are inverses means that f(g(x)) = g(f(x))= x.

In this case, if we take g(x) to be differentiation and f(x) to be integration as defined in the problem, we get f(g(1)) = 0.
f being onto is equivalent to f having a right inverse ie there exists a function g such that f○g=id.

Similar f being 1 to 1 is equivalent to f having a left inverse, ie there exists g such that g○f=id

So f being invertible is equivalent to f having a left and right inverse. It is easy to prove that in this case, the left and the right inverse are the same function (ie the inverse). I recommend to prove all these as an exercise.

Because integral○d/dx is not the identity, these are not inverses of each others