How do we know that we found all solutions of a differential equation? I hope that's not an extremely stupid question, but it' been in my mind since I was taught how to solve differential equations in secondary school, and I've never been able to find an answer.

For example, take the simple differential equation $$y'=y.$$

While I understand that $y=C\cdot e^x$ satisfies this equation, I do not understand why there can't be other solutions as well. So, how do we know this?
When you study the topic in University, you will probably encounter uniqueness theorems that show that for certain ODE's, not only does a solution exist, but this solution is also unique - there are no other solutions.

For first order ODE's such as that in your example, the relevant theorem is the Picard–Lindelöf theorem, which shows that your solution is indeed unique around a given interval. These intervals can then be strung together to show this is the only solution across the entire domain.

I'll add that for some other forms of differential equations it is not at all clear that solutions are unique. The most famous example is probably the Navier-Stokes equations in three dimensions, which govern the motion of gases and liquids all around us, and yet no proof of uniqueness in all cases is known.
In the case of y'=y...
Move along the x axis, follow the ordinate line up to the curve where the slope ='s the y-ordinate. In this way you trace out the curve, being only one curve save for translation.