Language models like GPT output a large vector of probabilities for the next token. Then a sampler decides which of those tokens to pick.

The simplest algorithm for getting good quality output is to just always pick the highest probability token.

If you want more creativity, maybe you pick randomly among the top 5 highest probability tokens or something. There are a lot of methods.

All that grammar-constrained decoding does is zero out the probability of any token that would violate the grammar.