Are Epsilon-NFAs a way to make designing NFAs easier?
An epsilon-NFA is a mathematical formalization.  Mathematics doesn't have a purpose, per se.

Why do we use this formalization?  Sometimes, because it is convenient.  Yes, it sometimes makes designing certain NFAs less onerous, and that's useful.  So, yes, your suggested rationale/motivation is reasonable.  It also makes some algorithms (e.g., Thompson's construction) simpler and easier to understand.  It's also intellectually interesting to know what power epsilon-transitions add.  There are multiple reasons why one might study this concept; different people might have different reasons for studying it at any point in time.  See also Why is non-determinism useful concept?.

What is "happening" when we execute an epsilon-transition?  Well, a NFA is not a physically implementable machine; it is an abstract concept.  So it is hard to map it to physical, real-world intuitions.  See also Why NFA is called Non-deterministic?.