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This is proved for the Chow ring in Fulton's "Intersection Theory", Section 16.1.

In topology it can be proved using the following facts (which it may be easier to find references for individually). Let $X$ and $Y$ be smooth manifolds.

1. A proper embedding $e: M\hookrightarrow Y$ of codimension $k$ with oriented (stable) normal bundle represents a cohomology class $[e]\in H^k(Y;\mathbb{Z})$. Call such a class representable.
2. Every element of $H^2(Y;\mathbb{Z})$ is representable.
3. Given a map $f: X\to Y$, the induced map $f^\ast: H^\ast(Y;\mathbb{Z})\to H^\ast(X;\mathbb{Z})$ is given on representable classes by pulling back a representing embedding by a smooth transverse map homotopic to $f$.
4. If $g:X\to Y$ is a proper map with oriented stable normal bundle, the pushforward $g_\ast: H^\ast(X;\mathbb{Z})\to H^\ast(Y;\mathbb{Z})$ is given on representable classes by composing a representative with $g$.
5. If $[f]\in H^k(Y;\mathbb{Z})$ and $[e]\in H^\ell(Y;\mathbb{Z})$ are represented by submanifolds, then their product $[f]\cdot [e]\in H^{k+\ell}(X; \mathbb{Z})$ equals $f_\ast f^\ast[e]$.

The formula you give follows on noting that $\Gamma_\varphi:X\to X\times Y$ satisfies $p_1\circ\Gamma_\varphi = \operatorname{id}_X$ and $p_2\circ\Gamma_\varphi = \varphi$.