When modelling a real world event by assuming it has probability p, what are we saying/assuming about how that event behaves?

This is a (good!) philosophical question not a mathematical one. In that respect it is not different from other questions related to the applicability of maths to the real world. Examples for such other questions are the physical reality of real numbers or the nature of limiting processes and infinity. And as always for those questions there will remain a certain gap which philosophy can't really close between the clearcut world of mathematics and messy reality. So do not expect final answers!

First, I suggest not to get too hung up about sets of measure zero for continuous distributions. If you are willing to accept the derivative in the concept of speed as a limit process of finite differences, you can as well focus on finite probability spaces without sets of measure zero and then take limits.

Next, you write "Most books I read interpret this as a long term relative frequency". This is a pity since there are a few more types of interpretation. The Stanford Encyclopedia of Philosophy states in their article "Interpretations of probability" among others subjective and propensity approaches. With these you do not need an infinite amount of independent replications to postulate something.

I will try to explain those two approaches in the most simple setting, the fair coin. In the subjective approach you argue that you believe (for whatever reasons) that one side of the coin is like the other, which is why you assign equal measure (of credibility) to the two events head and tails. In the propensity approach, you argue that the physical properties of the coin are such that both sides and hence both outcomes heads and tails are symmetric.

Subjective interpretations are great because you can assign probabilities to past events (What is the probability that the bus was on time yesterday?) or analyse deterministic computer experiments in a probabilistic fashion. The physical interpretation is nice because it creates a direct connection between maths and reality. In both approaches you end up with probabilities of "fifty-fifty" for the fair coin without requiring independent repetition of experiments.

Now, to answer your question:

"If you have real world events, and say that you model the real world, and assign probabilities to those events, what are you actually saying then?"

In the subjective approach you are actually saying nothing about the real world. You are only saying something about your subjective believes about the real world. In the propensity approach you postulate properties of a physical system (here: symmetry of the sides of a fair coin).

I think that the study of probability begins with our astonishing ability to imagine many different possible futures. Some of those futures are in some sense "likely" and some of those futures are "unlikely," but these concepts are rather vague and depend on our other astonishing ability to recall the events of the past. Depending on the accuracy of our memories, certain futures will "surprise" us if they occur and other future events will be met with a resigned attitude of "that's just what I expected."

This is all very vague and we want to find a better way to describe the "likelihood" of possible future events. So we begin to develop a measure of probability, whatever that is.

Some future events can be put to the test in a scientific way, with repeatable experiments. So I can, for example, roll dice and toss coins repeatedly to measure what happens. I can develop a theoretical approach to calculating probabilities for such events, using the idea of the number of possible outcomes. This leads to a belief in the measure of probability for certain simple types of events.

We then try to to extend our vocabulary to other kinds of events. This is where, in my opinion, probability theory starts to make some very extreme demands on our belief system. We are called to believe that non-repeatable events behave in the same way as repeatable events and we hope that our calculations that so far have been shown to be valid for repeatable events are also valid for discussing one-off events.

More deeply, we aren't really sure if the universe is deterministic or stochastic. If stochastic, the probability theory is probably a good model. If deterministic, then perhaps probability theory is not helpful.

The ancient Greeks had it both ways. The universe was governed by the gods (deterministic) but the gods were capricious and unpredictable, to the extent that they would decide the course of the future with the roll of dice (stochastic). This is where we get the phrase "it's in the lap of the gods" because they rolled their dice onto their laps...

Interestingly, even if the universe is deterministic, it may be so hard for us to asses all the variables required to predict the future that we are better off pretending that it is a deterministic universe after all.

Arthur C Clarke said that any sufficiently advanced technology is indistinguishable from magic. Perhaps the determinism of the gods only appears like blind chance to us...