I posted this question regarding intuition of denseness not too long ago, so maybe I can shed some light on your question by explaining this from my point of view.
If I wanted a definition of denseness, for example, I could read my analysis text or go on Wikipedia and be slapped with a bunch of jargon I might not know a priori. And if you read the question, I do have an idea regarding the definition of denseness.
But definitions aren't the most useful things when it comes to real problem-solving. Problem solving requires intuition, which is literally defined as "being able to solve something without conscious reasoning."
So, for example, let's take the line integral over a scalar field. You can go up to your average multivariable calculus student and give them a line integral problem and they'll bash out some algebra and calculus and give you an answer. But if you ask them, for example, "Can you quickly give me a nontrivial case where the line integral equates to zero," they might fumble with it. It's not because they're not thinking of cases quickly enough, but because they don't have intuition for what a line integral means.
Then, show them this animation of how to calculate a line integral. To be completely honest, this animation doesn't really help me solve any single line integral problem - that's what the definition and simplification formulas are for. But what this animation does do is show me what exactly is a line integral.
If you ask them the same "line integral equating to zero" question after showing them the animation, I think they'll be able to answer it much more easily.
tl;dr: giving someone intuition for a topic is explaining the topic to them in a way that doesn't help solve any single problem, but helps develop reasoning for how to solve problems. I think this might be a subset of "can you explain ___ to me," but I also think it's a really important specification.
Just as a side anecdote: I was once asked if you have a central circle $A$ of radius $r$, how many other nonoverlapping circles of radius $r$ can you place such that they are each tangent to $A$? To visualize it, I pulled out a box of Cocoa Puffs and picked a few up with my spoon, and quickly realized that the answer had to be $6$. I gained intuition for the problem by doing this, and then rigorously proved it not too long after.