Some other comment mentiones how it makes Frodo more influential and intimidating as they approach Mordor, allowing him to control Gollum with just his presence and voice. This isn’t very well portrayed in the movies. There is also mention of how a bunch of Orcs are scared off at the sight of Sam’s shadow when he is carrying the ring, as it appears to them as the shadow of a powerful elf lord.

As with other magic in Tolkiens universe, it is very diffuse. It grants the user great power, but the details of how it does so are very hard to pin down. We only get subtle hints.