As I understand it, one primary use of affine connections is to "connect" tangent spaces. Suppose I take a velocity vector $\dot{\gamma}(t_0)$ on a curve and at some point $\dot{\gamma}(t)$ also on that curve,where $t = t_0 +h$. One might then consider taking the derivative
$\lim_{h\rightarrow 0} \frac{ \dot{\gamma}(t_0+h) - \dot{\gamma}(t_0)} {h}$
But apparently this doesn't work since nearby tangent spaces are different spaces. So we need an affine connection to "connect" nearby tangent spaces to permit differentiation of vector fields.
I provided a definition of an affine connection here:
Breaking down what the definition of an affine connection says
My Question
How exactly does the definition of an affine connection allow me to then differentiate vector fields? I don't understand how an affine connection improves what we know about the relationship between the tangent spaces to permit differentiation.