OK, I'd try to remember that hearing things through an intermediary is often jumbled, and not focus on being told "I'm not allowed to see you again unless you meet my girlfriend" since there's the possibility that it was "Hey, I feel totally uncomfortable with you liking them so much, I'd really like to meet them before you go out again". Because that intermediary can often shorten things for convenience and give the totally wrong idea.
Me? I'd meet them. Maybe it works out great, maybe you realize they are a possessive insecure paranoid freak and you decide to leave the relationship you are in that seemed fine a week ago because it turns out there's veto power you weren't aware of. Maybe you make a new best friend, that you keep even if you break up with the partner. If you meet them you're much less likely to look back on it and wonder what if...
Feel free to reach out to them in email, don't expect your partner to do the best job of introducing you two. Get it over with, putting off nerve-wracking stuff usually just results in a lot of nerves over nothing. Don't feel afraid to ask to meet her alone if you want, or all together if you'd prefer it - for drinks, for coffee, for a long chatty dinner, at a mall, the zoo...whatever would make you comfy. Suggest a couple things and let her know you're open to alternatives. Get lots of sleep the night before. Remember she's as nervous as you, and once you meet there's a 96% (no really, that's a scientific fact I just made up) that both she and you realize there wasn't anything to panic about.
Happiness will never come to those who fail to appreciate what they already have.