Sessions in PHP allow information to be stored and accessed across multiple pages, solving the problem that HTTP is stateless and web servers do not inherently know who a user is or what they have done. Session variables can be used to store user information like username, preferences, etc. and last until the user closes the browser by default. The session_start() function must be called to start the session and then session variables can be set, accessed, modified, and destroyed through the $_SESSION superglobal array.