Recall that the Cech-to-derived functor spectral sequence is constructed as follows. We start with a sheaf $F$ and an open cover $\mathfrak{U}$. Then we can write the Cech resolution of the sheaf; take an injective (or Godement or...) resolution thereof to get a double complex. Let $C^{\ast,\ast}$ be the resulting complex of global sections and take the filtration $F^i=\bigoplus C^{\geq i,\ast}$. See e.g. Godement, Th\'eorie des faisceaux, 5.2. The rows of the $E_1$ sheet are precisely the Cech cochain complexes constructed from the open cover $\mathfrak{U}$ and the presheaves $U\mapsto H^i(U,F)$ (see Godement, ibid, just before theorem 5.2.4).

If $\mathfrak{U}$ has just two elements, $U$ and $U''$, then the $E_1$ term has two columns, the 0-th and the 1-st ones. Applying e.g. theorem 4.6.1 from Godement, ibid, one gets the long exact sequence

$$\cdots\to E_1^{1,i-1}\to H^i(X,F)\to E_1^{0,i}\to E^{1,i}_1\to\cdots$$

where the last arrow is the $d_1$ differential, $E_1^{1,j}=H^j(U'\cap U'',F)$ and $E_1^{0,j}=H^j(U',F)\oplus H^j(U'',F)$.