The key here is to focus on why you feeling anxiety.
Performance anxiety is something that is hardwired into our psyches. It comes from the natural fight or flight reaction. Think about the cave man that meets another cave man... he isn't sure if the other cave man is going to like him or try to kill him. His body reacts to the situation by being ready to fight or run away. So the key here is you don't know if the person you are performing for is going like your magic or not like them (in essence "kill" your magic ego). The two things that reduce that reaction is to increase confidence and to initiate a positive feedback loop.
Let's talk about increasing confidence. Many comments said to go out and perform more and some even suggested to fail. The more successful performances you have, the more confidence you have and the more you feel that people are going to react favorably to you magic. Thus, you've reduced the fear that they will kill you magic ego. If you fail and realize that your magic ego really isn't dead but "tis only a flesh wound" this reduces your fear of failure.
However, you also can increase confidence and reduce fail of failure by practice and rehearsal. Practice is perfecting the mechanics of an effect. Rehearsal is performing the effect repeatedly under performance conditions. That means doing the effect while giving your presentation while pretending you are doing it for an actual audience. It also helps to perform for a captive audience like you girlfriend, magic buddy, wife, dog, etc.
Scripting can also help. I know a lot of folks say to improvise, but then you end up just, as Eugene Burger said, narrating the adventures of the props. It becomes say-do-see patter where you say what are going to do, do it and then tell the audience to see the results. Start with strong magic and develop a strong presentation. A well scripted presentation comes across naturally. You don't have to think about what you are going to say. One lest thing to worry about. Also, having rehearsed the script with the effect provides a sense of security once you start.
In preparing the script, structure it so that you get a verbal reaction from your spectators before you begin the effect. Close-up magic is a conversation. That verbal reaction starts a positive feedback loop. You say something, they react (maybe by answering a question) and you react to their reaction. This gives you the sense of "hey, they like me" or, more importantly, "hey, they aren't going to kill my magic ego." In my parlor performances, I structure the first couple of lines in my presentation to draw the audience in and to get them to react. The reaction sometimes is a smile, sometimes it is laughter, sometimes it is a look of attention showing interest in what will come next. That reaction and engagement provides me with a positive energy.
One other way to get a positive feedback loop is to use something self-working for the first effect. That makes it easier on you and once you perform the first effect, you have built up a good enough rapport with the audience so that you can perform the second effect without fear of them killing your magic ego.