In essence, when requested, if you do not deny a fact within a specified time period (it varies between 10 and 30 days depending on your jurisdiction) it is automatically deemed admitted. But what does “deemed admitted” actually mean and how should you get the Court to do this?
It seems that you should not have to do anything more if the other side does nothing. But some judges and courts disagree. So the lawyer may choose to file a Motion to have the Admissions recognized by the Court. This can, however, tip off the other lawyers and the judge may give them more time to Answer. Under the Rules, as I am familiar with them, judges shouldn’t do this. But they do. Generally, what I do is make the Motion to the Judge. That way, if I encounter a judge who asks did I seek a court order that they be admitted (which absolutely should NOT be required per the Rule, but which nonetheless has happened) I have an answer. You need to do a risk-reward analysis on this, though, to decide which way to go.
If the lawyer for the other side has failed to answer Interrogatories or respond to Document Requests, (or the responses are inadequate) one way to get things admitted is to include the request in a motion to compel. This provides a little cover, hopefully downplaying a bit the significance of what you’re requesting as to the Requests to Admit.