It does not.
Modal logic is just predicate logic, except for the addition of a meaningless ‘box operator’, which, by stipulation, ranges over ‘possible worlds.’
Trouble is—the concept of a ‘possible world’ is the very thing that ‘modal logic’ is supposed to help us understand.
Which means it doesn’t help us understand anything.
It would be like somebody inventing a ‘coolness’ logic by adding a ‘coolness operator’ to predicate logic—without defining that operator, except by saying that it ranges over ‘cool people’ or ‘cool worlds’ or whatnot.
From a strictly formal viewpoint, there is absolutely zero difference between plain old predicate logic and modal logic. In other words, the two are absolutely identical——in other words, unless the box operator is question beggingly defined to range over all ‘possible worlds.’
All of the theorems and proofs relating to completeness and consistency are taken straight from predicate logic, with a few purely procedural changes relating to the presence of the ‘box’ operator (which, as previously stated, is either redundant or question-begging).
The problems that it solves are self-generated. Kripke frames are useless. Lindenbaum’s Lemma is useless. There is no substance there.
When people tell you that it helps illuminate ‘algebraic structure’ or ‘relational algebra’ or whatever, know that they are talking out of their hat.
It is a complete sham.