What does the dx *actually* mean mathematically and how come we can move it around?

Just finished calculus II with great success and I'm half ashamed to admit that I don't really know how all of the algebra with the dx's (what are these called anyways?) works. For instance, say we're doing a u substitution to solve an integral. Well, one would clearly do something like "Let u = f(x) → du/dx = f'(x) or du = f'(x) dx." Now, it's not immediately obvious to me how we can just move the dx to the RHS. This gets even more confusing with separable differential equations where whole problem depends on you separating these dx's; again, how the heck can we separate these in the first place?

It's just so odd because no one actually introduced the dx thing as an object; the only thing that was introduced properly was that d/dx is an ?operator? you apply to functions to get the derivative.

And then there's all of the volume/area formulas where the teachers say stuff like "let's take an infintessimal slice of this, then clearly...blah blah blah" I understand the intuition behind it but again, since I don't have a precise definition for what this dx thing is, I don't understand this stuff 100%.

It's just frustrating to me how I don't know this.