The proof aims to find a contradiction. You have to understand what the contradiction derived is, in order to understand why P is used as an input to itself. The contradiction is, informally: if we have a machine H(a, b) that decides “a accepts b”, then we can construct a machine that accepts machines that do not accept themselves. (Read that a few times until you get it.) The machine shown in the picture – let’s call it M – M(P)= does P not accept ⟨P⟩?
The contradiction happens when you ask: does M accept ⟨M⟩? Try to work out the two options to see how there is a contradiction.
M accepts ⟨M⟩ if and only if M does not accept ⟨M⟩; this is clearly a contradiction.
This is why it is essential for the proof to run P on itself not some arbitrary input. This is a common theme in impossibility proofs known as diagonal arguments.