5 ideas
An act is formalized as a mapping that assigns a consequence to every possible state of the world, so choosing an act means choosing which outcome you get in each state. Separating states (what the world does, beyond your control), consequences (what happens to you), and acts (the bridge you pick) lets both beliefs and values be read off from choices alone.
If you would prefer act f to act g whether or not event B occurs, you should prefer f to g even when you don't know whether B occurs. Equivalently, states where two acts yield identical outcomes should be irrelevant to choosing between them, which is what makes preferences separable across events and forces expected-utility form.
From a small set of consistency axioms on preferences among acts, it follows that a coherent agent behaves as if they hold a unique finitely additive probability over states and a utility over consequences, and maximize expected utility. Probability is not assumed as an objective frequency but recovered as a numerical summary of how a person is willing to bet.
Probability measures a particular person's confidence in a proposition, revealed by which bets they would accept, rather than a property of the world or a limiting frequency. This makes one-off events like a single election or a hypothesis being true legitimate objects of probability, and it treats disagreement between reasonable people as expected, not as an error.
Every real decision is modeled in a 'small world' that coarsens states and treats complex outcomes as simple consequences, because modeling the full 'grand world' of every contingency is impossible. The practical test of a small-world model is whether decisions made in it agree with those the grand world would dictate, and Savage acknowledges this consistency is not guaranteed.