I think the clue lies in a comparison with role-playing games, whether that would be video games or tabletop. There is a defining feature of role-playing game that has spread to many other games: The idea that there is a repeatable core activity, usually combat, which is embedded in a non-repeatable story and progression part of the game. The repeating part is something built upon strong game mechanics, and is fun even after playing it many times. And the story is interesting, has meaningful decisions, and brings some sort of progression or change to the repeating element. Ideally, there is synergy, the combination is more interesting than the sum of its parts.

Now of course there are a bunch of campaign board games that aren't labeled as legacy games. I talked about Roll Player Adventures, where we just finished adventure 8 out of 11. Once we finish adventure 11, we could completely reset the game and start over. But probably we won't, not until we have forgotten much about the story, or we decide to play with a different group. Roll Player Adventures is also one of many campaign board games that comes to mind (Tainted Grail would be another) in which the story is really great and makes you want to play, while the repeating core game mechanics aren't quite as good, and suffer from balance problems over the campaign.

In comparison, a game like Risk doesn't normally have a campaign or story at all, but has rather strong game mechanics. Some people play Risk over and over, because the randomness of the dice rolls, and the players making different decisions each time, provide already some variety. The same can be said about the Pandemic board game, or regular Clank!. But people apparently do get bored with the always same board of these games: You can get Risk boards with various themes, from Star Wars to Game of Thrones; there are different variants of Pandemic, including a recent World of Warcraft one; and Clank! has versions with sunken treasure, mummy's curses, or adventures in space.

Risk Legacy, Pandemic Legacy, or Clank! Legacy take a board game with strong game mechanics, and add a campaign mode to it. Clank! Legacy actually has a story book, but there are story elements to Risk Legacy and Pandemic Legacy as well. In every case there is some sense of progression, the game becoming a bit more complicated over time, with more options. And there is some modification of the board, adding new elements to consider, new locations, new pathways. It isn't about destroying game material, but adding new material to the base game. At the end of the campaign, you can't play the campaign again; but you do have a sort of "plus" version of the underlying game which can be repeatably played. The cards that you destroy are usually those that told you how to modify the game, and you don't need those anymore.

I haven't looked in detail into all the other legacy games out there. But there are several others that like the examples above are based on existing core games, and just add a campaign to it. And those that aren't, like Seafall, have been described as being less sound on the game mechanics side. A thoroughly tested base game seems to be a good starting point to making a legacy game.