Very hand wavy explanation
If we have two magnets they experience a force, either attracting them together or repelling them. Well the way they explain this in science is that both magnets emit a force carrying particle (a boson) and when this boson hits the magnet this is what makes it experience the force which we can observe.
Well the higgs boson is the same as the boson described above, but instead of magnetic force it's what makes things experience weight basically.
As for what it proves, it basically proves that the model that they use to describe things is right.
edit - I'm sure if you google it there will be much better, easier explanations.
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