If you have a vector field $\vec{A} = A_i \hat{e}^i$ and take it's derivative $\partial_j \vec{A} = \partial_j A^i \hat{e}_i + A_i \partial_j \hat{e_i}$ you see we took the derivative of the basis vector too, basic calculus ignores this, but I want to express the vector $ \partial_j \hat{e_i}$ in terms of my original basis so I say $ \partial_j \hat{e_i} = \sum_k \Gamma ^k _{ij} \hat{e}_k$ and get
$\partial_j \vec{A} = \partial_j A^i \hat{e}_i + A^i \Gamma ^k _{ij} \hat{e}_k$, the $\Gamma ^k _{ij}$ are 'connection coefficients' and we're obviously gonna have to use a 1-form acting on basis…