What does “put a floor under the crash” mean?

"We" and "us" refer to the Republicans; "he" and "him" refer to President Obama. This would be clearer if the quote was punctuated differently:

In Tampa the Republican argument against the President’s re-election was pretty simple: “We left him a total mess, he hasn’t finished cleaning it up yet, so fire him, and put us back in.”

Reporters probably didn't want to include quotes, because nobody actually uttered those words. That was just Clinton's paraphrase of the theme of the Republican convention, and, as would be expected, it's a very partisan paraphrase.

During their convention, the Republicans hammered home the point that the economy hasn't improved very much. Clinton is essentially making the argument that the economy is in bad shape, because (a) it could not have been expected to improve any more than it has, and (b) it was the Republican's fault that the economy was so bad to begin with.

As a historical footnote, Ronald Reagan successfully campaigned for reelection in 1984 by asking the American public "Are you better off than you were four years ago?" Clinton used that same line on his road to reelection in 1996. That line won't resonate well with Americans in 2012, so Clinton is trying to push the blame for that fact back an extra four years, to when the Republicans held the White House.

As for “put a floor under the crash,” that is Clinton trying to praise Obama's handling of what many regard as a weak economy. He is saying something along the lines of, “Sure, the economy is bad, but it would be worse if not for Barack Obama's leadership and policies.”

No, “put a floor under the crash” is not a common idiom.

The reference is to the Democratic position that Obama inherited an economy that was going over a cliff. The likely crash was at the bottom of a long fall. President Clinton was suggesting that the extent of that crash was significantly limited. Obama put a "floor under the crash" at a level that was much higher than it would have been had Obama not acted the way he did.

[This is not an analyis of what President Obama did, but rather an interpretation of what President Clinton said he did.]