Fun fact: YOU DON'T!
You do not handle/deal with them...as a friend it is your job to be there for them and reassure them that everything is OK.
Your best friend suffers from this illness called "anxiety", contrary to popular opinion: THEY CAN NOT HELP IT! They are sick, and while most people are on medication for it, some aren't even on medication and just go through life anxious all.the.time.
Your best friend does not need you to say, "calm down, you are overreacting," or "quit being a drama queen," stuff like that. This does not make it better, this makes them feel like they are unable to come to you if they're feeling bad or are having an episode; if you make them feel like this, you have failed at your best friend position and should consider yourself fired.
If he/she is acting differently because of it or being angry towards you, you definitely should mention it to them and let them know that it is not okay to treat you like that; but also follow up with, "what's going on?" "Are you okay?" "Let's talk about it," LET THEM KNOW YOU ARE THERE FOR THEM! If they do not have you there to lean on, they don't have anything.
Help them, reassure them that it will get better, it is not a big deal and they can get through it. Stop being rude and notice there's a problem and BE THERE!