The ability of abused illegally present women to be able to escape violent domestic relationship and get a visa was one of the hot button issues in the legislation. These individuals have had this right for several years, so we're not talking about something new, but there are many who would not want to "reward" these lawbreakers no matter how horrific their individual situations. Fortunately, public shaming pressured Republican leaders to allow for a vote on the bill in the House (the Democratic Senate was not a problem) and enough moderate Republicans supported the measure for it to pass. Some wise GOP elder statesmen (and women) presumably figured out that the one-two punch of angering women and Hispanics was probably not going to help get the Republican brand out of the dumps.
Which raises a broader question for the Republicans. Do you think it helps your image when you appear to be dragging your feet on immigration, VAWA and other issues only to eventually give in or does it make more sense to step out front and actually lead boldly on these issues? The Center for Immigration Studies is probably right when they say Latino voters probably won't switch to the GOP even if immigration reform passes. But that's probably because they sense that you don't really like them and are doing only what you need to do to survive as a viable national party. The rhetoric matters. How you deal with people in your party who regularly trash immigrants matters. What's in your national platform matters. What your presidential candidates say in debates matters. Which candidates you allocate your party's funds to matters. In short, just passing a bill alone won't cut it.