Let me begin by stating the theorem, then I'll give the outline of the proof.
Part 1: Set-up/observations
Part 2: Show $\Phi$ is well defined
Part 3: Show $\Phi$ is a group homomorphism
Part 4: Show $\Phi$ is surjective
Part 5: Show $\Phi$ is injective (Note: parts 4 and 5 require two lemmas whose proofs we will defer until part 6)
Part 6: Prove the two lemmas used in parts 4 and 5
Let's get started!
The following video illustrates the above rather nicely, especially the animation from 0:45 to 1:05. (After the 1:05 mark, the animation relates to covering spaces which we are not discussing in today's post.)