From what I understand, if a Jewish woman is not given a get by her husband, then she is still considered married in more conservative Jewish communities. This means she cannot remarry within that community, and any children she has with another partner will be considered illegitimate by them. The same goes for a man whose wife refuses to accept a get from him, although it seems this is a less common scenario (and even if a wife does refuse to accept the get, the husband can get around it with one of these, while the wife has no such option).
In this New Jersey case, the parties agreed they would first present their divorce to a rabbinical court and abide by that judgment. That court granted the divorce but left the decision of the get to the couple. When the couple went to “real” court, however, the judge ordered the husband to provide the wife with a get.
By doing that, the divorce court forced him to perform a religious action to which he had not previously consented. In other words, the husband and wife consented to do whatever the rabbinical court decided, but then the divorce court added an additional religious requirement (the get). The appellate court in New Jersey then decided that this amounted to attempted coercion of religious activity and was therefore unconstitutional.
This is the reasonable outcome, as judicial coercion of a religious act has been held to violate the Establishment clause. Note that the incorporation of the rabbinical court order is not constitutionally problematic because the parties had already agreed to abide by that order: the civil court was enforcing a contract, not coercing religious action.
Even if we ignore Supreme Court law, I still think this is the best outcome, mostly because it would be a huge mess if the courts got involved in people’s religious rites and ceremonies in this way.
But if I put myself in the shoes of the woman who is being denied the religious divorce: OH MY GOSH THIS WOULD DRIVE ME CRAZY. As the wiki page and the Volokh article explain, sometimes husbands use the denial of the get as a bargaining chip: “Give me the kids and I’ll give you the get,” for example. (Really classy, guys.) But can we blame them? The husbands are, after all, just taking advantage of a bargaining tool provided to them by their religion. If I were going through a divorce –- or any sort of dispute, really -– I would probably take all the (legal) advantages I could get.
The truly frustrating thing to me is that the religious law itself sets men up in an advantaged position. My (somewhat limited) research suggests that men who refuse the get often receive social sanctions, which is great, but… I would really want some legal sanctions too, if I were in that situation. Wouldn’t you?