V + -려 means intending to/just about to + V (i.e. 떠나려 - just about to leave) and can be combined with 하다 to mean "do with the intention of" which loosely translates to try. V + 보다 can mean to try, as you said, in a sense. It is an indirect form of imperative (indirectness is a common theme in spoken Korean). Rather than directly commanding the person to do something, you are saying "try this," but just because it is indirect does not mean it is always necessarily polite. There are a lot of other endings that end in 봐 though (-ㄴ가봐, -ㄹ까봐, -나봐 etc.,) so you may be hearing those as well.