I'll offer a perspective on this, that isn't exactly following the book's argument. Broadly speaking, it does not benefit the working class in any way to vote conservative; government regulation is required to restrict businesses and protect workers' rights. So, in order to gain votes, conservatives will often employ the tactic of publicising one particular issue that they know they are likely to be able to campaign well on, and trying to ensure that they win based off that issue. This works well, because if your candidate gets in, they are then able to vote on a whole raft of issues that the electorate may not support.

Previously in America, racial divides had been the basis of this tactic. Up until FDR, the Democrats had used divisions between black and white working class farmers to win the South (an interesting historical sidenote on this is the racial solidarity in the Populist Party, a third party that grew out of farmers unions in the south, and was eventually undermined by the democrats choosing a candidate who ceded to some of their economic demands). However, once the parties start to pivot, and especially once JFK/LBJ start to endorse the civil rights campaign, suddenly it's not as electorally viable to openly use racial divides as your campaign strategy. So, to keep your party relevant and to be able to stall civil rights legislation, you have to a) make your anti-civil rights operations covert (Google cointelpro), and b) find a new campaign issue to get your candidates in off the back of. That issue, this book posits (and I think quite rightly) was abortion.

Hopefully that's a somewhat clear explanation of the basic logic behind the explanation; if you're interested I can point in the direction of sources to read/watch/listen further.