Literally speaking, House Republicans aren't actually ending the debt limit hostage crisis; they are substituting one ransom note (demanding one dollar of cuts for every dollar of debt limit increase) for another ransom note (demanding that no member of Congress get paid if the House and Senate fail to pass pieces of paper stamped with the phrase "Budget Resolution"). But those ransom notes are so wildly different that there really is no comparison. In any practical sense, it's utter surrender.
Their original ransom demand was directly tied to the debt limit: the debt limit could only go up by an amount equal to any cuts enacted into law—no cuts, no debt limit. Their new ransom demand has nothing to do with the debt limit, doesn't require passing any additional legislation beyond raising the debt limit itself, makes no demands about cuts, and has an enforcement mechanism (cutting Congressional pay) that isn't even Constitutional.
Republicans would say that their demand is that the Senate pass a budget resolution, which is essentially a set of guidelines for the appropriations process. The reason Republicans are so excited about this demand is that they've convinced themselves that the Democratic Senate hasn't passed a budget in four years, and now they want to force Senate Democrats to finally pass a budget.
It turns out this is really more of a semantic argument than anything else. The Senate hasn't always passed a standalone budget resolution, but sometimes instead used actual spending bills themselves to serve the functions of the budget resolution. For example, the Budget Control Act of 2011 (the result of the previous debt limit fiasco and the creator of sequestration) serves as the Senate's budget resolution for the current fiscal year.
And budget resolutions don't even have the force of law. Remember the Ryan budget? Well, the House passed that. But nothing came of it. So to the extent Republicans are whining about the Senate not passing separate budget resolutions, they are basically just trying to create an issue that excites their base—it's not a huge substantive policy issue because the budget resolution guidelines are already in place. (Amusing side note: budget resolutions can contain reconciliation instructions, and as Republicans would probably like to forget, reconciliation allows Senate votes that can sidestep the filibuster—the vote for final passage of Obamacare being exhibit A.)
But even if you set aside the whole question of the budget resolution, House Republicans are saying they want to enforce their demand by cutting Congressional pay if no budget is delivered within three months. That's a lovely soundbite but a stupid idea, largely because it's unconstitutional (you can't change congressional pay in the middle of a legislative session). The key thing though is that it has nothing to do with the debt limit. Let's say the the House gets its demand, and the Senate or the House (or both) fail to produce a budget. The net result would at worst be a court fight over congressional pay.
It's true that extending the debt limit for just three months means Republicans will need to raise it again in three months time, if not sooner, but if they want to keep on taking votes to raise the debt limit, that's not the end of the world. The key point is that they have given up the debt limit hostage as leverage for anything that matters. Yes, they are still demanding a ransom, but it's like they've gone from demanding all the gold at Fort Knox to demanding a $5 gift certificate to Glenn Beck's favorite online gold reseller.
One last thing: I'm not saying House Democrats should vote for this. The Republican demand is still stupid, and if they care about it so much, they should get the votes together to pass it through the House. If they want House Democrats to give them a helping hand, they should put a clean bill on the floor. That's exactly what House Minority Leader Nancy Pelosi is saying. As to what Senate Democrats should do, there's really no point in speculating until the House passes something specific, but I'm fairly confident Senate Majority Leader Harry Reid won't accept legislation from the House containing a substantive ransom demand, even a procedural one.