Let $S, T$ be operators in $\mathcal{L}(V)$, the space of all linear maps from $V$ to itself. In my lecture notes, I have the definition of similar: "We say that operators $S,T \in \mathcal{L}(V)$ are similar if there exists an isomorphism (in this context of linear maps, a bijective linear map) $P$ such that $T = P^{-1}SP$. (If $V$ is $\mathbb{F}^{n}$, coordinate space, we call P a change of coordinates.)"
Could someone please try to explain this idea of "similarity" without invoking matrices or bases (I apologize in advance if I am asking for too much here)? Specifically, I am having some trouble really understanding the action of $P^{-1}$ on $SP$. What does it mean conceptually for two linear operators to be similar?
Sure, there are a couple different ways to think about this. One pretty general way is like this. Say you have two objects $X$ and $Y$ and an isomorphism between them. That isomorphism translates anything you do on one object to the other. So if you turn, or stretch $X$, what happens to $Y$?
Think for example of a mirror. If you raise your right hand, your image raises it's left hand. That's what's going on.
I think I'm being too hand wavy and missing an important point about your question. The point is that two operators are similar if one of them looks the same as the other except one looks like you did it directly to $Y$ and the other looks like you did it to $X$ and just watched what happened to $Y$.