Let $V$ be a smooth vector field along a curve $c$. The covariant derivative of $V$ along $c$ at the point $c(t)$ us given by
$\frac{D_cV}{dt}(c(t))=\lim_{s\to t}\frac{P_{c(t),c(s)}V_{c(s)}-V_{c(t)}}{s-t},$ where we define $P_{c(t),c(s)}$ to be the parallel transport of the tangent vector $V_{c(s)}$ to a tangent vector at the point $v(t)$.
Geometrically, the covariant derivative $\frac{D_cV}{dt}(c(t))$ represents the projection of $dV/dt$ onto the tangent plane of the surface, but I can't prove it clearly, how to give a intuitive way to see this?
Here we consider $c$ as a curve from a surface in $\mathbb{R}^3$