For applications in physics, maybe best to think of the ambient category as being one of certain geometric spaces.
For instance if the ambient category is that of smooth manifolds, then a category internal to that consists of a smooth manifold of objects and a smooth manifold of morphisms, such that all operations (source, target, identity, composition) are smooth functions between these.
This is well known in the case that the category happens to be a groupoid (the case that all morphisms are invertible). A groupoid internal to the category of smooth manifolds is just a Lie groupoid, the groupoid-generalization of a Lie group.
In physics the best well known example of this is maybe the concept of an orbifold: an orbifold is a Lie groupoid, hence an internal groupoid, hence an internal category (internal in smooth manifolds). This really just means that on the collection of all the points and that of all orbifold transformations between them ("twisted sectors") there is suitable smooth structure, and composition of orbifold transformations etc. is a smooth operation. For exposition of how this works in detail see Moerdijk 02.
If you have other examples of internalization contexts in mind for which you would like to gain more intuition, let me know what your context is and what motivates you, and I'll further expand in that direction.