D&D and the Adventurers League are made to be played with 5 players, that's the ideal number that Wizards of the Coast recommends in all of the D&D Adventurers League adventures. Playing with too many more (or less) player than the recommended number can lead to diminished fun for the players and DMs. Here are a few examples:
1) Too many players leads to each player having less time in the spotlight. Imagine a 2 hour adventure with 9 other players plus the DM, each player would speak on average only 12 minutes in the 2 hours and that doesn't count the DM talking more than a given player, which they probably do.
2) Combat starts to take quite a bit longer, since having more player characters means you'd need to adjust the combats by adding XP, thus more monsters causing more DM overhead to manage the combat. Each round is now taking longer and some decisions will take longer.
3) Having more players means people are going to want to be heard and decisions in game are going to take more time, again taking longer.
4) Many AL games are played at stores or conventions with strict time slots. 7 players was the threshold that was chosen to still be able to run the game in the given time period. Even 7 is pushing it for most tables.
Maybe as a DM you can manage all of these things, but it ultimately makes for a sub-optimal experience and it's not something the D&D Adventurers League wants to promote, especially if there are new players.
Telling someone there's no room is hard, but when I do that, I do it for my own sanity as well as the enjoyment of the other players at the table.
Source: I am the community manager of the D&D Adventurers League, Robert Adducci