The question is, which votes should be counted in a union representation election? The House wants to revert to an old rule in which all workers eligible to vote are counted as having done so. If they didn't actually cast a vote, they are recorded as a "no" vote. That means that joining a union does not just require getting a majority of the votes cast, but a majority of all workers. It's as if defeating George W. Bush would have required a majority of all voting-age Americans, regardless of voter turnout; as Joan McCarter has written, this provision codifies vote fraud.
That's what the House wants to shut down the FAA over. Since they're taking the position that air traffic controllers would be kept on as essential employees, they probably figure that public anger would be minimal even as they cause some pain, including furloughs for other workers and loss of revenue as airlines stop collecting ticket taxes. But House Republicans aren't just going out of their way to be dicks to workers: they've proposed a short-term extension that eliminates federal subsidies to 13 rural airports, including ones in the home states of the 3 Democratic senators with the most authority over the bill. And they're not pretending that's a coincidence; the chair of the House Transportation and Infrastructure Committee referred to it as "just a tool to try to motivate some action to get this resolved"—the "this" being passage of the anti-union provision.
According to a letter from Rep. Nick Rahall (D-WV) to John Boehner, the series of short-term extensions has already had costs such as slowing airport construction projects. But, you know, anything to make it harder for workers to join unions.
Update: From the White House:
The Administration strongly supports passage of a clean extension of Federal Aviation Administration (FAA) programs, as the Congress has done 20 times without controversy, in order to allow bipartisan, bicameral negotiations to continue on a full reauthorization.
H.R. 2553 includes controversial provisions that, because they have not been negotiated, needlessly threaten critical FAA programs and jeopardize thousands of public and private sector jobs. Without timely passage of a clean extension, all of FAA’s capital accounts (Grants-in-Aid for Airports, Facilities and Equipment, and Research, Engineering, and Development) would be shut down, and approximately 4,000 employees would be furloughed. FAA’s ability to award new grants, including for infrastructure upgrades at airports across the country, as well as to move forward with vital testing and implementation of the Next Generation air traffic control system, would come to a stop.