The verb 'figure' on its own isn't used very often. When it is, it means to suppose or imagine. For example, 'I figured you'd be tired when you got home'. This usage is more typical of American English.
'Figure out' means to deduce or calculate, and is similar to 'work out'. If there's a puzzle, a mystery, or a problem of some kind, you need to figure out the answer or the solution.