Why Uniqueness Explains Method Of Images

Why does uniqueness of Poisson equation guarantee that method of imaging works?

If say, we find a distribution that matches the boundary potential (thus matches the boundary condition) we still need to make sure that our potential that arises from this distribution still satisfies Poisson equation to make sure that we found the solution to the boundary value problem, however for us to be able to do this we need to know the charge density, which we don’t actually know do we? So then how can we check that our "trick" actually gave us the sought for potential, since we cant actually verify it satisfies Poisson equation.

I.e we would like to check: $nabla^2V=-frac{rho}{epsilon_0}$ but how do we know $rho$ if we don’t know the charge distribution?