Let $\pi:E\longrightarrow M$ a smooth fibre bundle. A connection is a linear bundle homomorphism $\Phi:TE\longrightarrow TE$ such that $\Phi$ is a projection to the vertical bundle $VE\subset TE$.

I read that a connection in $E$ is equivalent to a section $\Gamma:E \longrightarrow J^1E$, and the space of connection is an affine space.

What is this relationship, explicitly? and why it's an affine space?

Thanks