While I haven't played Diablo 2, I can tell you any% speedruns are speedruns with the goal of finishing the game as fast as possible, while allowing the player to ignore side content. The percent symbol refers to the % of activities completed in the game - in this case it refers to not completing side activities in the speedrun. Other speedrunners attempt to complete all activities available in the game, known as a 100% Speedrun.
Without further qualifiers, "any%" usually also means all (or most) glitches and skips are allowed. So you're free to ignore a lot of the "main" content, as long as you're able to reach the end of the game. In case of multiple endings, any is OK, "bad ending" too.