"The" solution is not entirely correct. Every ODE has an infinite family of solutions, usually parameterized by a constant of integration:
is a more general solution. Even works; i.e., the function satisfies the differential equation from the book.
To try to understand the domain where the solution makes sense intuitively, try thinking of solutions as curves (i.e., integral curves) in the -plane, and think about the uniqueness of solutions. Given any point where , you can can find a constant such that the curve
passes through the point . However what happens at ? For any (other) constant , the curve , , is also a solution to the DE, and it meets the solution at the origin. Uniqueness is violated.
For any point in the domain , there is a unique solution of the DE passing through that point that is defined for all . The same is true for the domain . This is why the authors exclude .
Hope that helps!