weather是天气，天气好坏固然关系到每个人，keep a weather eye out原来指的是在海上张帆远航时头等重要的大事，也就是时时注意观测风云变幻，引申为注意观察。

【例句】
Let's keep a weather eye out for those guys across the street. If they cut their prices or offer more dishes than we do, we need to be ready to fight back as quick as we can before they grab any more of our customers.
让我们注意观察街对面的那些家伙。如果他们降价或提供比我们更多的菜肴，我们需要准备好在他们抢走我们更多的顾客之前尽快反击。

I think your roommate is athief; so keep your weather eye out around him.
我认为你的室友是个小偷，在他身边你可得保持警惕。