Use this phrase when you're trying to make a decision, and you haven't completely decided yet. The phrase "lean toward ___" means to slightly favor one choice, but not be completely decided yet. For example, when there's an election coming up, you can ask a friend:
Who are you leaning toward?
Whether to use "toward" or "towards" is a topic that many English grammar teachers get upset about, but it's not really important. Use "toward" in formal writing, but in spoken English you can use either one.