For any covering $\mathcal U = \{U_\alpha\}_{\alpha\in A}$ of $X$, we can form its Cech nerve $C(\mathcal U)$, an abstract simplicial complex with vertex set $A$ whose simplices are exactly the finite subsets $I\subset A$ such that $\bigcap_{\alpha\in I} U_\alpha\neq\emptyset$. We can then glue standard simplices together as specified by this abstract simplicial complex to get a CW complex $|C(\mathcal U)|$. If $\mathcal U$ is a good cover, i.e. all finite intersections are empty or contractible, and $X$ is sufficiently nice, e.g. a manifold, there is a homotopy equivalence $|C(\mathcal U)|\to M$ constructed by mapping the $0$-simplex corresponding to $\alpha$ to some point of $U_\alpha$ and then iteratively extending it over simplices (which is always possible and unique up to homotopy by contractibility). The cellular complex of $|C(\mathcal U)|$ is precisely the Cech complex of $U$. Thus the Cech cocycle $\omega_{\alpha\beta\gamma}$ is just a representative of the pullback of $\omega$ in the cellular complex. Integration is pairing against the fundamental class, which is an element of $H_2(X)\cong H_2(|C(\mathcal U)|)$, so it can also be expressed as a sum of formal elements corresponding to nonempty intersections of $U_\alpha$'s.

In fact, for special coverings one can identify this class explicitly: Choose a triangulation of $X$, i.e. an abstract simplicial complex $K$ and a homeomorphism $|K|\cong X$. To every vertex $x$ of $K$ corresponds a contractible open subset of $|K|$, the star of $x$, which is the union of the interiors of all simplices which have $x$ as a vertex (together with $x$ itself). Together, these form a good cover of $|K|$, and the corresponding Cech nerve is just $K$ itself. The fundamental class is given as the sum of all top-dimensional (in this case $2$-dimensional) simplices, with signs corresponding to the orientations. Thus pairing with the fundamental class sends a Cech cocycle $\{\omega_{xyz}\}$ to the sum over all triangles $t$ of $K$, with vertices $x_t,y_t,z_t$ in this order, of $\omega_{x_t,y_t,z_t}$.