Thanks, DKW! I remember watching the game with my dad, who got really upset about it. Not that our team (the Falcons) was involved, but because it was so weird and unaccustomed.
As Kurt is known as a HUGE Christian, I would have loved to have seen him in the post game interview do a bit by comic Jeff Stilson “We would have won the game…until Jesus made me fumble”.
You know, that’s a damn good question, and I don’t know enough about the rules to give a definitive answer. Of course that doesn’t prevent me from venturing a guess 
a) The Rams thought they were declining a different penalty (remember, there were flags everywhere, and the officials did a huddle before they called offsides). Maybe, as Freddie the Pig points out, they thought it was encroachment rather than offsides.
b) The decision to put time back on the clock meant the Rams got a do-over on their decision to accept or decline the penalty.
c) Since the entire thing happened between plays, the Rams had the right to change their minds before the ball was spotted
d) The players may have said “decline” while the coaches were saying “accept.” I don’t know who wins out in that scenario.
I welcome any alien NFL rule overlord who can explain it.
My 13-year-old NFL rule book (they don’t publish it every year) states the following:
Note that the onus is on the referee to explain the consequences of accepting or declining. If he makes a mistake, as they sometimes do (being human) in complicated cases, the logical follow-up is to correct it and poll the captain again, not to stand on the earlier (improperly framed) choice.
A free kick on an untimed extension down is an extremely unusual play; most likely neither the refs nor St. Louis thought about it beforehand. Ordinarily an offsides penalty on the last play would be a no-brainer: the Rams are ahead, they decline it, and the game’s over. No need to even ask the captain. Just announce it, wave it off, and leave the field.
Except that, in this one particular case, the game isn’t over. The Cardinals mention it to the refs, the refs explain it to the Rams . . . and give them the chance to accept the penalty and call a kneeldown.
By the way, it had to be offsides and not encroachment. A dead-ball foul like encroachment would wash out the play, and the free kick scenario wouldn’t be an option whether the penalty was accepted or not.
And according to this story that’s pretty much the way it happened. Thanks FtP