"odds" comes from betting. It refers to the ratio of what you bet to how much you win, which is decided by how likely each result is.
If something is probable, we say "the odds are in its favor". If something is not probable, "the odds are against it".
When a movie says the odds are stacked against him, it means that things are set up to make it very unlikely he will succeed. He is likely to fail, because so many things about his situation are stacked up against his success. If a gambler bet on him, the odds would be against him.