I've read this online, but I haven't seen this proved in Munkres Topology. Has it been? If so, where?
In any case, here is my attempt to show it using the tools given in the book. Please verify.
The book has 2 exercises on contractible spaces. One says contractible spaces are path connected and the other says a space is contractible if and only if the space has the same homotopy type of a one-point space. Since simply connected is defined as path connected and trivial fundamental group, I think that if a space has the same homotopy type as that of a one-point space, then the fundamental group of the space is trivial. Is that right? If so, I can work out the details myself.
Definition of fundamental group
Definition of simply connected
The context is the claim made here in Example 70.1