"Politically vital independents": Here in the US, general elections are not decided as much by Democrats and Republicans as by Independents. This is because a large percentage of those registered to a specific political party will vote for their own party and will support their own party policies no matter what happens. To a certain extent, their votes can be taken for granted during a general election (as opposed to in a primary election). However, the power to decide general elections lies with independents because, in theory, they can be persuaded to vote for either party. They represent about 20% of the electorate - and if Your political party can capture the indenpendent vote -!you win! Therefore, to win an election and retain support for a policy, such as the Affordable Care Act (aka Obamacare), it is vital to have the support of independent voters.
If hostility to the Affordable Care Act has "shot up" then the Republicans have an opportunity to use opposition to the act as an election issue. It means that, in the opinion of the author, there is a much higher chance of the act being changed or completely replaced because the political will to maintain the policy has evaporated.