The Phantom Menace ends with Obi-Wan Kenobi defeating Darth Maul, Naboo being freed, Palpatine becoming Chancellor and Obi-Wan agreeing to train Anakin Skywalker. Qui-Gon dies of his wounds, the droid army is disabled, and the Federation is defeated. The film closes with a celebration on Naboo.
The finale runs on three fronts at once. The Gungans join Queen Amidala against the Federation's droid army, Anakin sits hidden in a starfighter, and Qui-Gon and Obi-Wan fight Darth Maul in Theed. The film cuts between them, and the front that settles the war is the least planned one: Anakin accidentally destroys the droid control ship, which shuts the whole army down. The Jedi duel decides who lives, but Anakin's blunder decides who wins.
The quiet beat around the duel matters most, because Palpatine becoming Chancellor sets up his rise to power. One reading (opinion) is that the real winner is Palpatine, since Sidious's hand never appears in the final celebration. The Federation's defeat removes a pawn, while the Senate stalling on Coruscant has already handed Palpatine the office he wanted.
The second thread is Obi-Wan agreeing to train Anakin. The Council had hesitated because of the boy's age and fear, so the choice rests on a promise to a dying master rather than on the Council's trust. Anakin's destruction of the droid control ship from inside a starfighter is an accident, not a planned act, and the film leaves open what he is. Whether he is the prophesied Chosen One is Qui-Gon's belief, not something the ending confirms.
There is production context behind that restraint. According to the Wikipedia entry on the film, Lucas began writing the prequel trilogy on November 1, 1994, under the working title The Beginning, and Anakin was first written as twelve years old before being reduced to nine. In my opinion, a nine-year-old at the center of the finale sharpens the Council's worry about age and fear, because the boy who saves Naboo is also the one they doubt.
A comparison with the rest of the saga explains the shape. This is the fourth film in release order, arriving after A New Hope, The Empire Strikes Back and Return of the Jedi, but the first in story order. Its ending therefore works as a setup rather than a closure: a celebration on Naboo, a new Chancellor and a new apprentice. You can follow both threads into Attack of the Clones and Revenge of the Sith.