I am currently enrolled in a General Relativity course, and was taught about the connection but I can't really wrap my head around it qualitatively. All I can think of is that it must have something to do with the coordinate system that one uses to describe a space(or space-time) but I can't give it a geometric interpretation**(Check the 2nd EDIT)**.
EDIT 1: I am searching for an explanation in terms of the curvature of the space and the coordinates.
EDIT 2: Upon searching for an answer, I found that the relation of the connection with the covariant derivative offers some insight: The connection term in the covariant derivative is an extra term to the normal derivative that is there in order to account for the changes in the coordinate basis vectors. If anybody could use this type of logic to give a complete geometric interpretation of the connection, it would be great!