I truly don't get it. Empathy is “code” for judicial activism? “Judicial activism” is obviously code for the judging of Democratic judges. What “empathy” is obviously code for is a group-identity-based nomination, in particular the nomination of a woman. And what's wrong with that?
One of the deepest problems in philosophy is the relationship between general rules and their application in particular instances. Rules don't apply themselves. And there can be no infinte regress of rules that tell us how to apply rules. Judgment is completely unavoidable. And, hey, maybe there's a reason we call judges “judges”!
Anyway, suppose there's a fact of the matter about the external totality of facts that constitute a “situation.” (This is a problematic supposition; it's not so easy to individuate “situations”– but suppose.) This totality can't possibly fit in anyone's head, so the situation has to be edited by selective awareness. So we're left with a perception of the situation as edited by habits of attention. Within one's perception of the situation, certain features will stand out as especially salient. This can be moral or legal salience or something else, depending on the purposes of judgment. One's history is obviously relevant to the habits one brings to attention and to the recognition of features of a situation relevant to judgment. One way a rule may be misapplied is to fail to recognize the relevance of an objective feature of the situation to the rule. This can happen at the initial step of editing — one doesn't notice the feature at all — or at the step of the judgment of salience or relevance — one doesn't see why it matters.
Even when there is unanimity about the meaning of a rule, there may be disagreement about whether and how a rule applies in a particular instance due to differences in the habits of attention and sentiment that guide judgment. If a society has a history of inequality, people within different groups may have developed very developed very different but also very reasonable habits, and will therefore make very different judgments, for good reasons, even if there is zero disagreement in the abstract meaning of rules.
The Supreme Court is a deliberative body. If it is extremely homogenous in composition, there's a good chance that judgment will become biased in the direction of the characteristic habits of the largest group. Difference in ideology provide some check, but they may also paper over subtle an not-so-subtle uniform assumptions that lurk behind political and methodological disagreement. Some of this sort of disagreement may be over the relevance of things like empathy to legal judgment in constitutional cases, but surely the capacity to put oneself in the position of parties involved in a case before the court is relevant. And the ability to put oneself in the position of another is certainly improved if one has at some point been in a similar position. (E.g., No man was ever a teenage girl; white people rarely face the kind of subtle discrimination routinely experienced by even privileged black people; etc.) One plausible understanding of “empathy” in this context is simply a heightened sensitivity to features of certain kinds of cases that are missed or downplayed due to the habits of mind and sentiment common to most current judges.
I think that, other things equal, the Court would improve the quality of its judgment by including more women and minorities. However, other things aren't equal. My sense is that the best-qualified women and minorities are likely to have substantive views about Constitutional interpretation I disagree with. So I'm likely to be unhappy with Obama's nominess because of ideology. But holding ideology fixed, I think there's a strong reason to prefer a well-qualified woman to a well-qualified man. And I think another woman would likely increase the scope of empathy on the court in a pretty straightforward and desirable sense.
Anyway, I'm just being dense. The Republicans were going to attack basically any Obama nominee as a monster of judicial activism anyway, and so they used the unusual and thus salient appearance of the word “empathy” to get started. It's stupid, but politics is stupid.