Kentaro Tomono's answer is partially correct. In this case, "everything (they've got)" is the direct object, and "it" is an indirect object. The sentence:
They gave it everything they've got.
could be reworded as:
They gave everything they've got to it.
(although most people wouldn't actually say it that way)
In this case, the implied "it" is actually "the game" being discussed, so this is equivalent to:
They gave the game everything they've got.
Saying the sentence with and without "it" are both OK, and are roughly interchangeable, though without the "it" the feeling is a little more vague, and is basically saying "they expended a lot of effort", whereas with the "it" it is emphasizing that all the effort they expended was definitely focussed on trying to win the game, and not anything else.