Are Epsilon-NFAs a way to make designing NFAs easier? It seems that their purpose is in making designing certain NFAs less onerous? Does this mean that no automata that are implemented have Epsilon-transitions, and that they are kind of a mathematical conceit we adopt for our own ease of design? I'm having a lot of trouble understanding what is "happening" when an Epsilon-transition occurs - is it a way of saying, "have the NFA now switch, automatically, to this next state"?

EDIT: I'm  a CS student taking CS Theory for the very first time, so apologies if this question is very rudimentary. I'm a narrative person and I often understand abstract ideas through examples or by trying to understand the problems the ideas are engaging with. I ask very basic questions, but those questions help me hook into the ideas I'm studying.
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?.