Role of connection in Lagrangian of a field theory

I understand some of the basics of differential geometry. I have two questions about gauge theory from the physical perspective:

• I understand the connection is literally a "connection" between fibers, a horizontal tangent space, to provide the parallel transport of a lifted point to its endpoint based on the curve it gets lifted from. I know this is also a 1 form on the fibres themselves, and takes the value of a Lie algebra when these fibres are principle fibers. In what way is this connection a Lagrangian? What physical features do they share?

• Following this, the parallel transport is the integral of a connection. I think I am clear on why this is, because of covariant derivatives and whatnot, but what physical intuition do we have for this being the action?

Obviously this is in a classical field, else we would have action $to$ path integral – I think – and I would accept either a general answer or a specific case study. For example, in an electromagnetic field.