Picturing the how the covariant derivative acts on vector fields
Solution 1:
Suppose that the manifold $M$ is embedded in $\mathbb R^n$. Then a vector $v \in T_p M$ at point $p \in M$ can be considered a vector in $\mathbb R^n$.
The covariant derivative $\nabla_u v$ for some $u \in T_p M$ then is the ordinary derivative of $v$ in the direction $u$ projected on $T_p M$.