How will a guy act around a girl he likes but who he thinks is "out of his league"?

Anonymous
I invited this guy to hang out sometime. He did not say no explicitly but he did pretty much turn me down. I won't get into a description of his behavior, but I'll just simply say that for quite a while now he has definitely acted like he a has a crush on me (usually somewhat in a shy way), to the point where other people were noticing and commenting on it to me. So it's not just in my head.

What throws me is that most of the time he tries to avoid contact with me, and when he DOES have to be near me, he gets pretty clearly nervous. Talking to me, he stumbles on his words and can't make eye contact and all of that. (But the smile when he does!! Man...) So here I am, figuring I (and others?) have read him wrong somehow. An unbiased guy friend listened to my super long story and said he definitely sounds like he has feelings for me, but that because he gets so nervous around me, he probably thinks i'm out of his league. If that's true, WHY on earth would be turn down an invite to hang with me? Wouldn't that make him happier? I don't understand this "league" idea.

Anyway. How might a guy act around a girl if he has a crush on her but thinks she is out of his league?
How will a guy act around a girl he likes but who he thinks is "out of his league"?
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