Perhaps someone from the States can explain something to me, seeing as I'm a Brit. So some law gets passed by both Congress and Senate, and the President then signs it into law, effectively creating a government body (I'm thinking the new Consumer Financial Protection Bureau, but I'm sure there are plenty of other examples). This has been considered to be a Good Thing (TM), and then Congress then de-funds it, effectively shutting it down.
Why don't they just repeal the original law that created it in the first place?
In the US, it is much, much easier to prevent something from happening in politics than it is to get something to happen. The don't just repeal it because they don't have the votes, so the most effective way to attack it is to attack the funding for it. To pass almost anything, it requires a simple majority in the House, a 60% majority in the Senate (to pass the inevitable filibuster), and the president to sign it. The other alternative is 2/3 majority in House, 2/3 majority in Senate to override presendential veto. The Consumer Financial Protection Bureau was passed when Democrats controlled the House, Senate, and White House. Currently, the Democrats still control the Senate (with less than 60% majority) and the White House, but Republicans control the House. Republicans want to repeal the law, but Democrats want to keep it. If Republicans tried to repeal it, it would be blocked by the Democratic Senate and White House. If they try to defund it, that will still be blocked by the Democratic Senate and White House, but since that is tied into the federal government's budget which funds everything, eventually something has to give. It ends up becoming a giant game of chicken usually and at the last minute, a compromise that no one likes very much is approved.