The electron exchanges a virtual photon with the positron without annihilating. The photon carries momentum $k = p - p'$ with $k^2 = t$. Dominates at small scattering angles (forward scattering).
The computational chain
From diagram to number—every step of the amplitude calculation.
The diagram
The electron and positron exchange a virtual photon without annihilating. The photon carries momentum $k = p - p'$ with $k^2 = t$.
Feynman rules expression
$$i\mathcal{M}_t = \frac{ie^2}{t}\left[\bar{u}(p')\gamma^\mu u(p)\right]\left[\bar{v}(q)\gamma_\mu v(q')\right]$$
The electron current $\bar{u}\gamma^\mu u$ and the positron current $\bar{v}\gamma_\mu v$ are connected by the photon propagator. Compare to the s-channel where the currents are $\bar{v}\gamma^\mu u$ (annihilation) and $\bar{u}\gamma_\mu v$ (creation).
Spin-averaged squared amplitude (massless limit)
$$\overline{|\mathcal{M}_t|^2} = \frac{2e^4(s^2+u^2)}{t^2}$$
The trace calculation is identical in structure to the s-channel with $s \leftrightarrow t$. The $1/t^2$ pole gives the characteristic forward peak of Coulomb scattering.