This might be my own bias, but I think differential geometry is a really natural area to study. When we look out at the world around us, we see lots of objects that seem smooth, but are not flat. As such, it is only natural to try to study geometry using techniques from calculus, which is the starting point of differential geometry. In this vein, many of the concepts in differential geometry come from real-world considerations.
Let me try to explain how this works with connections. Most people take it for granted that we have a consistent idea of which direction "East" is. That is to say, if you tell me "the sun rises in the East," we will both know which direction that is, even though we are standing in different locations. However, it turns out that formalizing this idea of consistent directions at different points requires some work, and it's not obvious how to do it.
After a fair amount of thought, it turns out to be equivalent to being able to compute the directional derivative of vector fields, because we can say that two vectors at different base points $p$ and $q$ are "the same" if the directional derivative is identically zero when we move from $p$ to $q$ along the shortest path* from $p$ to $q$ (i.e., a geodesic). And this is exactly what a connection $\nabla$ is: it is a way to compute the covariant derivatives of vector fields.
Now, there is an added complexity in that there are many possible connections and each induce their own notion of parallel transport (and also their own notion of what a geodesic is). However, the basic idea comes from a natural physical idea and the generalization is helpful because it appears in many other mathematical problems.
*Strictly speaking, the correspondence between the curves whose tangents are parallel with respect to $\nabla$ (i.e., $\nabla$-geodesics) and paths which minimize the total length only holds true for metric connections. For general connections, we use the $\nabla$-geodesics rather than the shortest paths to go from $p$ to $q$.