I have a few friends who suffer from anxiety as well and it took me a while to really understand this as well.
The short answer is, you can not convince them. The best you can do is to show them you like them, again and again. (I mean in a normal, day to day friend way. Not like, saying every few moments you like them.)
We all are a little anxious from time to time, but to them it is much more severe. They struggle with a continous doubt that what they are doing is bad in someway or that they might offend you and others by doing seemingly mundane things to the rest.
Because of that it is very hard for them to hold on to this feeling/ emotion of "This person actually likes me". They can succeed in that, but know that on a bad day, they will doubt it all again.
From personal experience I found that the best wat to handle it is to let them know you understand that they have this doubt every now and then. And that you will be there for them when you need them. Aside of that, just be a good friend.
I found that trying to help by suggesting things or urging them in a certain way to change about it is a bad move. It is very hard for them to change this and it will feel as if they are not okay or good enough for you and make things only worse.