So, when you apply the function to the first element in the set - e.g. Zero or Nil in the case of lists - you're actually testing to see the function works. Then in the inductive step you base everything on the assumption that p holds for some n and of course if that's true then p must hold for Succ n but you have to prove this by taking Succ from n and thus going bakc to its predecessor which is also the hypothesis p(n).
So, to reiterate
assumption: if hypothesis then conclusion
       if p(n) then p(Succ n)
proof of assumption if p(Succ n) = Succ(p(n)) then we've won. If pn+1) = p(n) + p(1) then we have liftoff!
I'm not going to go any further in case I'm once again on the wrong track.
Cheers
Paul