The crux is that insurance companies *want* to cover birth control. They really, really do. Birth control is incredibly cheap compared to pregnancy, delivery, and actual children.
Obama riled up Republicans on contraception, and then delivers a knock-out punch.
After two solid weeks of Republicans rapidly escalating attacks on contraception access under the banner of "religious freedom," Obama finally announced what the White House is proposing: an accommodation of religiously affiliated employers who don't want to offer birth control coverage as part of their insurance plans. In those situations, the insurance companies will have to reach out directly to employees and offer contraception coverage for free, without going through the employer. Insurance companies are down with the plan, because as Matt Yglesias explained at Moneybox, contraception actually saves insurance companies money, since it's cheaper than abortion and far cheaper than childbirth. Because the insurance companies have to reach out to employees directly, there's very little danger of women not getting coverage because they are unaware they're eligible.
That's the nitty-gritty. The fun part of this is that Obama just pulled a fast one on Republicans. He drew this out for two weeks, letting Republicans work themselves into a frenzy of anti-contraception rhetoric, all thinly disguised as concern for religious liberty, and then created a compromise that addressed their purported concerns but without actually reducing women's access to contraception, which is what this has always been about. (As Dana Goldstein reported in 2010, before the religious liberty gambit was brought up, the Catholic bishops were just demanding that women be denied access and told to abstain from sex instead.) With the fig leaf of religious liberty removed, Republicans are in a bad situation. They can either drop this and slink away knowing they've been punked, or they can double down. But in order to do so, they'll have to be more blatantly anti-contraception, a politically toxic move in a country where 99% of women have used contraception.