They feel pretty confident, because they gerrymandered the hell out of the country's electoral map in the 2010 redistricting. That's how they currently have a majority in the House: there is a significant overall preference among the voters for Democrats, but the electoral map has been drawn so as to concentrate Republican voters and break up blocs of Democratic voters, ensuring that they will have enough "safe" districts to retain control of the House at least until the 2020 redistricting.
There's nothing "hyperbolic" about it. The party that's in power during redistricting always does this. It so happens that this time around it was the Republicans, and they've set up the districts so that they feel safe in pursuing an unpopular agenda.
The actual hyperbolic conspiracy-theory angle on redistricting is that the long-term plan, funded by the Kochs and others, was to ensure that only the most extreme Republicans are safe, and that moderate Republicans will be in danger of primary challenges from extremists.
Political scientists have long said that there's not a lot of evidence for a change in macro outcomes due to gerrymandering. In most scenarios, if anything, gerrymandering makes sitting politicians on both sides more safe. It's really hard to actually tip the balance such that one side gets significantly more seats because if you spread your supporters out efficiently (51% your party) and then cluster the other party, you get a bunch of weakly held seats. However, if you want safe seats, you get fewer of them.
Esch state draws its own districts, so there were 50 different "party in power" scenarios in 2010. Democrats in California gerrymandered themselves into a super majority. In Texas Republicans drew themselves some very safe districts. But there is no national gerrymandering or "party in power".
GOP moderates will take a hit. The more conservative members that are driving this shutdown will end up strengthening their base, so they won't feel as much pressure.
In addition, some theorize that the Republicans can use the chaos and votes supporting the ACA to go after vulnerable Democratic senators.
And maybe I'm getting too cynical, but the extreme Republican line for years has been that government is the problem, not the solution. So I don't think the members chasing the base have the same interest in a functioning state that the rest of them do.
This is exactly the case they are in. If you look at it from the typical tea-party perspective, they were in a win-win situation. Either get concessions on AFA, or shut down the government, which in their view is the root of the problem to begin with. I have to think this strategy will end up backfiring on them eventually, especially when you consider the long term trends in political leanings in the country, but for now, they got exactly what they wanted.
I'm reminded of the dog that finally caught a car and then didn't know what to do with it.
