The people in my local D&D Adventurers League are all grown men who say I am too young to join, even though I am 13.
Is that true? Is there a minimum age to play in AL games?
Adventurers League is not supposed to have a minimum age requirement. When I was playing at local AL tables, one of the regulars was a 10-year old with autism and ADHD, and when he was focused he was good. When he couldn't focus we sent him on errands, which he was glad to do anyway because he'd been sitting still longer than he could really manage. The DM took over his character for a little bit while he ran a couple blocks to get sodas or whatever. Win-win.
The players may be trying to shoo you away because they've tried integrating younger players and had it go badly. Find out in what way it went badly and resolve not to be the kind of player that creates those problems. The DM is within their rights to ask you to leave if you are disruptive, so know your character's abilities and spells and so on and be a helpful part of the team.
With that said, having an official AL table means they can't actually turn you away without risking their AL status. They shouldn't be turning people away for reasons like that, only for cheating or repeated super-disruptive behavior. I've only seen one person ejected from a table and that was for threatening violence against another player.
Assuming your AL game is at a local gaming store, the official AL organizer is probably the store owner or manager, and might also be the DM but probably isn't aware the players or the DM are discouraging you from participating. Talk to them and let them gently remind the DM and others that this space is supposed to be for you too. While the players may have turned it into a private club for themselves, they are kind of misusing what they've been given. If they want a private club and they're grown men, they can pool their money for a folding table and play a home game in someone's basement.
According to their FAQ there doesn't seem to be any requirement for players other than having a DCI number. The only requirement to register a DCI number is to either be 13+ or have a parent/legal guardian's consent.
No, but GMs basically have the right to kick you out of the group if you are acting in a disruptive fashion. As a 13 year old, your brain hasn’t finished developing yet, and it’s entirely possible that you might act in a socially disruptive fashion without realising it or intending to, simply because the portion of the brain responsible for impulse control hasn’t fully developed in kids your age.
Additionally, depending on your location, there may be regulatory requirements for staff and volunteers that interact with children, like the Australian government’s requirement for Working With Children Checks and their associated state-level ID cards. If the event was intended for adults, it’s possible that the staff haven’t had these checks done. As a result, it’s entirely possible that it may not be legal for you to remain at the store without the presence of at least one of your parents or legal guardians.