The short version is that a legacy game is a game that is played as a campaign, where you finish the game multiple times over the life of the campaign. Legacy games are generally played with the same group of players and permanent changes will happen based on the result of each session. As you play this means the game will get a different feeling and at the end it can be drastically different than the start. You could end up altering or destroying various pieces and parts of the game.
As a note there are people who have set up instructions to make a copy of a legacy game replayable from the start.
Legacy games are designed to be played over the course of a campaign, usually with the same players, and permanently change over time. As such they have been compared to tabletop role-playing games like Dungeons & Dragons. New rules can be introduced as the campaign goes on, allowing for the game to expand both mechanically and thematically. Games can use the expanding campaign as a mode of storytelling; Pandemic Legacy: Season 1 uses a three-act structure to tell its story. Daviau describes legacy games as "experiential" in contrast to traditional games, which are "repeatable". He compared his legacy games to that of a concert where you "buy a ticket for an experience" while Haoran Un of Kotaku describes the idea as "avant-garde performance art".
Legacy games break certain covenants that players expect from traditional board games. Permanent, physical changes can occur to components based on game outcomes and player choices. For instance players might be instructed to write names on cards, place stickers on the game board, or destroy some components. This causes each copy of the game to be unique at the end and has earned the legacy genre criticism in that there is a finite amount of replayability. Some games have been designed to be replayable with refill packs or non-permanent stickers while others are still playable with the final permanent changes once the campaign is over.