Which is correct?
"Ich versuche, Menschen mit ähnlichen Hobbys zu treffen." "Ich versuche Menschen mit ähnlichen Hobbys treffen."
If both work, what is the difference? If one doesn't make sense, why?
You need zu. You have a conjugated verb that is not a modal (assisting) verb therefore the following verb needs to be in infinity. Example in English: I try TO learn Spanish, I try TO help people .... Zu takes the place of TO in English.
Because versuchen stands with infinitive.
It's been a long time since I've done German, so I don't 100% understand what all of this means. However, I can impart a bit of knowledge, I believe. It looks like "treffen" is being used as an infinitive, and imparting a bit of Dutch knowledge (the languages are quite similar), it seems to me that "zu" would trigger the infinitive and be more correct. I may be completely wrong, though.
You need "zu" here. That's just the way it is, and I don't think the question "why" has any sensible answer. For modal verbs or "lassen", you don't need "zu", e.g. "Ich will Menschen mit ähnlichen Hobbys treffen". I believe however that for a construction like this with any other verb, you need "zu".