The reasoning is that the derivative of an exponential is again an exponential. Further, in order to be able to cancel and so get 0, we must have functions that will cancel one another.Since in a differential equation with constant coefficients, the only "function" is the dependent variable, y, that means that the various derivatives must give the same "kind" of function- i.e. be an exponential.
If you are asking how we know the solution is an exponential, the answer is we don't. Trying an exponential leads to other solutions as well.
