'Better off' is a common idiom meaning 'to be in a better situation'. It's usually followed by the verb+ing. We prefer using 'better off' to simply saying 'better' in contexts like this. Here is another example: There has been an accident on the motorway. You'd be better off finding another route. 'with' is grammatically correct, but in informal language it's OK to omit it.

the sentence is not using good grammar. Sometimes words are dropped when being very casual. You would be better off alone than to remain with someone who is making you unhappy. To explain the phrase "Better off" on its own, I'd say better off means being well cared for, well fed, and/or to have your needs met.