How do you make sense of derived units??

Units have always stumped me. I understand the base units (meters, seconds, kilograms, etc.) and I understand what they actually measure (distance, time, mass, etc.), but I cannot for the life of me seem to wrap my head around what operations within units actually quantify.

Like what does multiplying units actually represent? I understand division to an extent, being like frequency per something, but even my understand of division with units is extremely iffy. And for exponentiation I am entirely clueless.
The Newton (kg * m/s²), for instance, measures force, which is mass times distance per some unit of time squared. But what does any of that actually quantify? I understand distance per some unit of time, and I barely understand distance per some unit of time squared, but I only understand it as being a simplification of distance per unit of time per unit of time. If you asked me what a unit of time squared represents when it's not in the denominator I would have absolutely no answer. And, to me, multiplied units raise even more questions. What does mass times acceleration actually mean? What is actually happening when I say "times"?
It's just always been such an elusive concept to me, and I think my lack of understanding on fundamental measurements and quantities is seriously prohibiting me from understanding other concepts. I didn't even think about the fact that I could research this until earlier this week, but I've had no luck so far.

(TLDR: what does multiplying or dividing units actually mean)