\[ 0 \to H^1_{dR}(X_t/K)^* \otimes \mathcal{O}_{X_t} \to \mathcal{E}\to \mathcal{O}_{X_t}\to 0 \]
These are just extensions for trivial by trivial (no connections on the first and third objects).
This encodes the Coleman integral structures on \(X_t\), as when you choose a basis \(\underline{\omega}=(\omega_i)\) for \(H^1_{dR}(X_t/K)\) (giving a basis for \(V\) is the same as giving a basis for \(V^*\)), define a connection on \(\mathcal{E}\) as
\[ \begin{pmatrix} 0 & \underline{\omega} \\ 0 & 0 \\ \end{pmatrix} \]
and the flat section represents integrations. Say \((0, 1)^t\) is a vector in the fibre of \(\mathcal{E}\) at point \(b\), then the flat section is \((\int_b^x \underline{\omega}, 1)^t\).
We want to find the global version of this,
the global version of de-Rham is the relative de-Rham with Gauss-Manin connection \((\mathcal{V}, \nabla)\)
\[ 0\to \pi^*(\mathcal{V}, \nabla)^* \to \mathcal{E}\to \mathcal{O}_X \to 0.\]
Whose fibrewise version is the above.
The connection defined on \(\mathcal{E}\) is given by
\[ \begin{pmatrix} \Theta & \pi^*\underline{\omega} \\ 0 & 0 \\ \end{pmatrix}\]
where \(\Theta\) is the dual (transpose) of the Gauss-Manin connection.
Recall that a connection \(\nabla: \mathcal{E}\to \Omega_X^1\otimes \mathcal{E}\) given by matrix \(d\cdot I_n + \Lambda\in \mathrm{Hom}(\mathcal{O}^n, (\Omega^1)^n)\)
For a connection to be flat, it means \(\nabla^2 : \mathcal{E}\to \Omega_X^2\otimes\mathcal{E}\) is zero, i.e. the curvature is zero.
In terms of matrix \(\nabla = d\cdot I_n + \Lambda\), as we have seen in coordinate form of connections, the flatness condition \(\nabla\circ \nabla=0\) expands to \(d\Lambda + \Lambda\wedge \Lambda = 0\).