The first half of the question has been answered in the comments, so let me address the second half of the question.

We want to define Hecke operators on the complex $R\Gamma(X, \omega^k)$, because this is the complex whose $H^0$ is modular forms for $k \ge 1$ (and whose $H^1$ is dual to cusp forms for $k \le 1$).

The definition of $T_p$ is going to be, essentially, the composite $$R\Gamma(X, \omega^k) \xrightarrow{(p_2)^*} R\Gamma(X_0(p), p_2^* \omega^k) \xrightarrow{???} R\Gamma(X_0(p), p_1^* \omega^k) \xrightarrow{(p_1)_*} R\Gamma(X, \omega^k).$$ (Some of these $*$-s should be $!$-s, I don't remember exactly which but the paper will tell you.) The middle arrow $???$ needs to be there: you can't just compose $(p_1)_*$ and $(p_2)^*$, the composition doesn't make sense. So this is why we want to make a map between $p_2^* \omega^k$ and $p_1^* \omega^k$.

As for how we get a map: there is a canonical isogeny $p_1^* E \to p_2^* E$, because this is what $X_0(p)$ parameterises; so we get a "naive $T_p$" by setting $???$ to be the pullback of differential forms along this isogeny (or the pushforward along the dual isogeny, if you prefer). The subtle thing is normalising this by the "correct" power of $p$, so that the map is defined integrally and its ordinary part is not trivially 0.

The first half of the question has been answered in the comments, so let me address the second half of the question.

We want to define Hecke operators on the complex $R\Gamma(X, \omega^k)$, because this is the complex whose $H^0$ is modular forms for $k \ge 1$ (and whose $H^1$ is dual to cusp forms for $k \le 1$).

The definition of $T_p$ is going to be, essentially, the composite $$R\Gamma(X, \omega^k) \xrightarrow{(p_2)^*} R\Gamma(X_0(p), p_2^* \omega^k) \xrightarrow{???} R\Gamma(X_0(p), p_1^* \omega^k) \xrightarrow{(p_1)_*} R\Gamma(X, \omega^k).$$ (Some of these $*$-s should be $!$-s, I don't remember exactly which but the paper will tell you.) The middle arrow $???$ needs to be there: you can't just compose $(p_1)_*$ and $(p_2)^*$, the composition doesn't make sense. So this is why we want to make a map between $p_2^* \omega^k$ and $p_1^* \omega^k$.

As for how we get a map: there is a canonical isogeny $p_1^* E \to p_2^* E$, because this is what $X_0(p)$ parameterises; so we get a "naive $T_p$" by setting $???$ to be the pullback of differential forms along this isogeny (or the pushforward along the dual isogeny, if you prefer). The subtle thing is normalising this by the "correct" power of $p$, so that the resulting correspondence is defined integrally and its ordinary part is not trivially 0.