We can see the vectors of the tangent space $T_pM$ to a smooth manifold as velocities of curves. This is elaborated here.
Each velocity $\gamma'(0)$ corresponds to a derivation $D_{\gamma}(f) = (f \circ \gamma)'(0)$ as seen in the Wikipedia article.
But why is every derivation also a velocity (vector)?
Once you choose local coordinates, a basis for the derivations is $\left\{\dfrac{\partial}{\partial x^1}\Big|_p,\dots, \dfrac{\partial}{\partial x^n}\Big|_p\right\}$. Then there's an obvious curve with tangent vector $\sum\limits_{i=1}^n a_i \dfrac{\partial}{\partial x^i}\Big|_p$.