很高兴认识你 is a general greeting, which is used when you meet a person for the first time. The literal translation is "Very glad to know you!". In practice, It's like saying "Nice to meet you!" in English. Nice to meet you! is shorthand for It's nice to meet you! in my opinion.
很高兴认识你 can be paraphrased as: 我很高兴认识你 (I'm happy to meet you.)or 认识你是(一件很让人）高兴的事情 (it's a very nice thing that I can meet you). Pulling 很高兴 ahead has some effect of emphasis. You can also say 认识你很高兴. With 我 implied, it sounds more like a generic formal greeting, rather than a personal one.
If you say 认识你我很高兴， that doesn't sound like a generic business meet, but rather personal(may have some romantic implication sometimes). It implies that you are impressed by the person you meet, you are kind of person I want to meet, and those kind of ideas.