I thought the radio exchange was with the other planes, not the tower.
Reagan airport is very busy. It is very close to DC (basically one crosses the bridge on the Potomac and ends up right on the Mole, with the White House and Congress right there) and civil airliners land and take off one after the other. I'm pretty sure civil traffic had been put on hold during the duration of the event. The pilot just directed the plane towards the runway and landed there, it all happened in a couple of minutes. I don't think the pilot would have done that if he didn't know in advance the runway was free for him to land. I beleive that in normal circumstances, after receiving the request for emergency landing, the tower informs the pilot that his/her way is clear and instructs him/her on which runway to land on. I didn't hear that happening here, or did I miss it?