Please tell me which sentence is correct and please give me your explanations.
1. She claims to have met the Prince.
2. She claims to meet the Prince.
1. is right. 2. isn't.
Meeting the prince happened in the past (or so she claims).
You could also say
"She claims that she has met the prince" (present perfect).
"She claims that she met the prince" (Simple past - some adverb of time would be expected here).