In a strange way, or strangely enough. It has two rather different jobs. First, it describes how something is done or how it seems: "the room was weirdly quiet", "he was acting weirdly". Second — and this is very common in speech — it sits at the front of a sentence to say that the whole fact that follows is surprising: "Weirdly, I slept better after only four hours." Used that way it means much the same as "strangely" or "oddly", and young people especially use it a lot. It is casual and friendly, so keep it out of formal writing and reports. Also notice that "weirdly" often carries no criticism at all — "weirdly good", "weirdly moving" — it just marks something as not what you expected.