Do they do anything? Or are they just to collect? I'm kinda confused.
Artifacts have several objectives tied to collecting some or all of them. So if you want to reach the maximum level and get that score multiplier up, you better start collecting.
After you collect them, you can then sell them for extra coins or gems.
Also, whenever you collect a full set, you get a special themed hat that you can put on your runners. Yes, even Montana Smith.
It's worth noting that you can only ever get 1 Artifact during a run, so it's best to intentionally end your run after you grab one. That way, you can get the next one sooner.