A grad school professor contacted me about research before I applied
This isn't very common in my field, but my field doesn't involve labs so I'm confident our fields are different. I can see why you would feel skeptical of what seems like, on the surface, such an important stroke. Always investigate things that seem too good to be true. Here's some steps I would take:
- Talk to your undergraduate advisors and letter writers. This is the most important one; knowing you, your work and your field, they can help you with your specific questions, and possibly suggest things to find out that you haven't even considered.
- Research the facility and professor. It sounds like you've already done this, or at least started on it.
- Ask the professor how your work came to his attention. You don't have to make it challenging, just tuck it in with a reply along the lines of "Thank you for contacting me, I'm very interested. Can you give me some more information about (research specifics)? Also, what brought my work to your attention that made you feel I would be a good fit for your lab?" The answer to this should help you answer your questions about whether he's seen your letters of rec or your application, or something else.
- If everything looks good and you feel you're a fit for this lab, don't sell yourself short! Most grad students go through some form of imposter syndrom, so this can actually be an obstacle you'll need to overcome.
Good luck with grad school regardless; applications can be stressful and a lot of work.
In my field it is fairly common for a prof to take the initiative. This will typically happen when (s)he got a grant for funding a number of PhD students and now needs to fill those positions on fairly short notice. So if (s)he hears of a promising candidate (s)he will likely contact that candidate.
Do a little research on the professor to see if you'd be interested in working with him. If so, accept his offer to visit his lab and see if you'd like working in that environment; especially if other grad students share the lab. If the professor agrees to sponsor you and your application is strong, it will make your acceptance more likely.