How do scientists use data at the crater site to estimate when a meteorite or asteroid impacted the earth?
Several possible ways..
If the crater has been buried, just date the first rocks that are on top of the crater but not disturbed by it; this can be done using fossil-based dating if no good radiometric techniques are available. This has the advantage of being unambiguous, but the problem could be, especially for land based craters, that there is a gap between crater formation and new sediment deposition.
If you can find a layer of ejecta (famously the layer of Iridium-rich ejecta from the K-T impact) and relate it to your crater, then you can often date this very precisely, more so than the disturbed area around the crater itself. This has the potential to be more accurate than any other method, but you have to be sure that the Iridium-rich layer you are dating actually corresponds to the crater.
A large impact will almost certainly melt some rock locally at the site of impact, resetting any radiometric 'clocks' in that rock. If you can locate this impact melt, then there are a variety of radiometric techniques that you can apply to it, depending on the mineralogy. Potassium-Argon would be an example.