Tolkien wasn't super clear about this and what hints he did put in, the movies left out, but the answer is: The Ring controls people.
In the book, Frodo does this to Gollum on the slopes of Mount Doom, and curses him to "yourself be thrown into the Fire" if he ever touches him again.
I feel Peter Jackson didn't understand that -- as he also didn't understand why Gollum, having sworn on the Ring not to harm Frodo, couldn't (personally) harm Frodo (and therefore had to lead him into Cirith Ungol for Shelob to kill him instead).
"Smeagol promised!" "Smeagol lied" was never how it worked.
Which is why Jackson had Frodo still apparently be drawn to the Ring after it was bitten off him and attack Gollum and push him over, because he thought Gollum just "randomly fell off the edge" in the book. It wasn't random, it was the effect of Frodo's curse using the Ring's power.
At all levels the Ring's power is to give its bearer power. For small mortal folks like Hobbits that means the gift of invisibility, which, like the Ring of Gyges in Plato's Republic, removes social consequences for one's actions. For people like Denethor or Aragorn it would have meant the power to command armies, which is what Sauron thought was happening when Aragorn marched on the Black Gate and why he sent every soldier he'd got to take it back. For Gandalf ("through me the Ring would wield a power too great and terrible to imagine") or Galadriel ("all shall love me and despair") it would have meant phenomenal power over nature.
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