Vector bundle with flat connection over simply connected manifold is trivial

Question

I am trying to finish a proof of the statement in the title. Let $M$ be a simply connected smooth manifold and $E \twoheadrightarrow M$ be a vector bundle with a flat connection. Since the bundle is flat we can choose a local trivialization at every point such that the connection is trivial in all trivializations (with null 1-form matrix). Because of this, and because of the transition formula between the connection matrices of 1-forms, the transition functions between such trivializations are constant. I would like to use this fact, together with the simple connectedness of $M$ to either reduce the structure group to the trivial group or to build a basis of global sections, but I am a bit rusty with these proofs and I cannot come up with an intelligent idea to do this. Any hint?