Anyone means any person at all, without specifying who. When a teacher asks “Can anyone answer this question?” she's inviting any single person in the class to respond. When a sign says “Anyone can learn to play piano,” it means the opportunity is open to every person, regardless of who they are.

The word emphasizes that it doesn't matter which person: anyone will do. If you're looking for someone to help carry boxes and you say “I need anyone who's available,” you're not being picky about who helps, you just need a person. This is different from everyone, which means all people together. If anyone can solve a puzzle, it means at least one person can do it. If everyone can solve it, it means all people can do it.

You'll often hear anyone in questions and negative statements: “Is anyone home?” or “I don't know anyone here.” In positive statements, we usually say someone instead: “Someone is at the door” sounds more natural than “Anyone is at the door.” But when you're emphasizing the openness of an opportunity or invitation, anyone also works in positive sentences: “Anyone can join our club.”