Atiyah Sequence and Connections on a Principal Bundle

First note that the adjoint bundle $ad(E_G)$ can be canonically identified with the vertical tangent bundle $V E_G / G$: send the pair $(p, \xi)$ consisting of a point $p \in E_G$ and a Lie algebra element $\xi$ to the value $p \cdot \xi$ at $p$ of the fundamental vector field generated by $\xi$. Moreover, the Atiyah bundle $At(E_G)$ is just a fancy way of writing $T E_G / G$.

Thus, a splitting of the Atiyah sequence is nothing else than a diffeomorphism of $T E_G / G$ with the direct sum $V E_G /G \oplus T M$ (taken over $M$). In this way, you recover the definition of a connection as a complement to $V E_G$. The horizontal bundle is the image of $TM$ under the above isomorphism $V E_G \oplus T M \to T E_G / G$. Alternatively, you can view a splitting as a projection onto $V E_G$ or as a $G$-equivariant lift $TM \to TE_G$. These equivalent viewpoints give you the definition of a connection as a connection $1$-form and as a horizontal lift operator, respectively. I've expanded a bit on the different but equivalent viewpoints in an answer to a different question.