This business of "contractions" is just an overly formal way of saying the following: "commute all ladder operators until they annihilate the vacuum; your matrix element is given by whatever delta functions you pick up along the way."

Commuting the $a$ in $\phi(x)$ with $a^\dagger$ in $| p \rangle$ produces a delta function, but $\phi$ integrates over the momentum, so you pick up the coefficient $e^{-ip.x}$ of $a$. (The factor $1/\sqrt{2 \omega_p} $ is cancelled by that in $| p \rangle$).

This business of "contractions" is just an overly formal way of saying the following: "commute all ladder operators until they annihilate the vacuum; your matrix element is given by whatever delta functions you pick up along the way."

Commuting the $a$ in $\phi(x)$ with $a^\dagger$ in $| p \rangle$ produces a delta function, but $\phi$ integrates over the momentum, so you pick up the coefficient $e^{-ip\cdot x}$ of $a$. (The factor $1/\sqrt{2 \omega_p} $ is cancelled by that in $| p \rangle$).