As several commenters have pointed out, there are multiple interpretations of probability. In my answer, I'll focus on the subjectivist interpretation.
"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? If you say that the chance that the bus will be on time have probability 0.3, what are you actually saying then?"
The short answer, for the subjectivist, is that you're saying something about your personal judgments. Exactly what you're saying depends on precisely what brand of subjectivism one endorses. What's interesting, and contra what g g has said, the issues here are not purely philosophical; there are mathematical results that shed light on your question. I turn to those now.
The only place to start is with De Finetti. According to De Finetti's subjectivist theory, there is some collection $\mathcal{B}$ of events of interest. An agent is to buy and sell various bets on these events. For each $B \in \mathcal{B}$, the agent's price function $p: \mathcal{B} \to \mathbb{R}$ gives the her fair price $p(B)$ for a bet that pays \$1 if $B$ occurs and nothing otherwise. A price function is said to be coherent if there is no way to buy and sell a collection of bets with our agent such that she is guaranteed a net loss. More precisely, $p$ is coherent if there do not exist $B_1,...,B_n \in \mathcal{B}$ and real numbers $d_1,...,d_n$ such that $$\sum_{i=1}^{n} d_i (\mathbf{1}_{B_i} - p(B_i)) < 0.$$ This simple set up is enough to state the very interesting
Theorem 1. Suppose that $\mathcal{B}$ is an algebra. Then $p$ is coherent if and only if it is a finitely additive probability measure.
Back to your questions: On this interpretation probabilities are coherent price functions. To say the probability that the bus is on time is 0.3 is to say that \$0.30 is your fair price for a bet that pays \$1 if the bus is on time.
Later authors working in the subjectivist tradition extended these ideas dramatically. Perhaps the most important work here is Leonard Savage's Foundations of Statistics. For Savage, the fundamental notion is not a price function, but a qualitative preference ordering over actions. For example, you might prefer taking the bus ($f$) to walking ($g$), and we write $g \precsim f$.
More precisely, Savage postulates a set of states of the world $\mathcal{S}$ and a set of outcomes $\mathcal{O}$. Actions are functions from $\mathcal{S}$ into $\mathcal{O}$. For example, suppose $s \in \mathcal{S}$ is a state in which the bus breaks down. Then the outcome $f(s)$ of taking the bus is that you're late to work. Now, as with De Finetti's notion of coherence, Savage imposes certain normative constraints, in the form of axioms, on the ordering $\precsim$. I won't list the axioms here, but, for example, $\precsim$ should be a weak order.
To get probabilities, we proceed in two steps. First it can be shown that if $\precsim$ satisfies the Savage axioms, then there exists a unique qualitative probability $\preccurlyeq$ on events in $\mathcal{B}$, defined to be an algebra of subsets of $\mathcal{S}$. The idea behind qualitative probability is the following. First, $\precsim$ induces an order on the set of outcomes $\mathcal{O}$ by identifying outcomes $o$ with constant actions $\hat o$, i.e. $\hat o(s) = o$ for all $s \in \mathcal{S}$. Continuing our example, suppose you prefer getting to work on time $o_2$ to begin late $o_1$, so $o_1 \precsim o_2$. Now consider two events, say the event $A_1$ that the bus driver is Terry Tao and the event $A_2$ that the bus driver is not a mathematician. How can we determine which event you regard as more probable given only your preferences $\precsim$? The idea is to look at two actions $f_1$ and $f_2$, say
$f_1$ returns $o_2$ if $A_1$ occurs and $o_1$ otherwise. That is, if you perform action 1, you're on time if Tao is driving and late otherwise.
$f_2$ returns $o_2$ if $A_2$ occurs and $o_1$ otherwise. That is, if you perform action 2, you're on time if the driver isn't a mathematician and late otherwise.
Suppose $f_1 \precsim f_2$. But since you prefer being on time $o_2$ to being late $o_1$, intuitively this means that you regard $A_2$ as more probable than $A_1$, i.e. $A_1 \preccurlyeq A_2$, because it's reasonable for you to prefer the action that makes your preferred outcome more probable. Using this idea to define the relation $\preccurlyeq$ we can show
Theorem 2. Let $\mathcal{B}$ be an algebra of subsets of $\mathcal{S}$. If the preference relation $\precsim$ over actions satisfies the Savage axioms, then $\preccurlyeq$ is a qualitative probability relation on $\mathcal{B}$, i.e. for $A,B,C \in \mathcal{B}$, $\preccurlyeq$ is a weak order with $\emptyset \preccurlyeq A$ and with $A \preccurlyeq B$ iff $A \cup C \preccurlyeq B \cup C$ when $A \cap C = B \cap C = \emptyset$.
The second step is to derive numerical probabilities. For this a few technical conditions are required. But assuming these technical conditions and the Savage axioms are all satisfied, we have
Theorem 3. There exists a unique, finitely additive probability measure $\mu$ on $\mathcal{B}$ that represents $\preccurlyeq$, i.e. for all $A, B \in \mathcal{B}$ we have $A \preccurlyeq B$ if and only if $\mu(A) \leq \mu(B)$.
This provides a rather remarkable answer to your question. The numerical probability 0.3 that you assign to the event that the bus is on time is a representation (in the precise sense given above) of your personal preferences. Subjective probability in the Savage framework derives from your ordinary preference for being on time as opposed to being late.
In both the De Finetti and Savage frameworks, we start with objects that are less mysterious than probability (e.g. betting behaviors or ordinary qualitative preferences) and show that probabilities can be derived from these objects. This gives precise meaning to the that claim I made at the beginning, namely that probabilities represent your personal judgments.
