The standard technique is "put yourself in their shoes," which works for me. However, this doesn't always work with people completely different from myself, because I can't imagine reacting the way they are.

My impression is that virtually all of empathy comes down to refining precisely this skill. I suspect that when we find there's someone whose perspective we cannot understand, we're encountering a limitation to our empathic strength just like our inability to lift a heavy weight indicates a limitation in our physical strength.

I think How to Win Friends and Influence People is fantastic in this regard. Carnegie's thesis is that all people have a deep desire to feel important, and giving them that feeling is to everyone's benefit.

Another resource I've found helpful is An Open Heart by the Dalai Lama. The farther you go into the book, the weirder it gets, concluding (if I recall correctly) with a description of how praying to infinitely many Buddhas from infinitely many parallel universes will help you in your pursuit of enlightenment. But before it gets really alien and weird with its Mahayana cosmology, it's really quite a pragmatic guide to becoming happier and more empathic through the cultivation of compassion.

I've personally gotten a great deal out of the Riso/Hudson system for the Enneagram. It describes nine personality types in a way that many people find insanely insightful (and in a falsifiable way, I should add). One of the most dramatic insights I got from working with this system was just how incredibly different other people really are, and in what ways. This helps a huge amount with understanding others' reactions when they're so completely alien to our own motivations. The main problem with the Enneagram is that it has a huge learning curve; usually certain people leap to mind as you read the detail descriptions of certain types, and one type in particular tends to give the reader an "I've been caught" sort of feeling, but to get more detailed insight than that into the types usually takes a long time - and trying really hard to get through that learning curve quickly can actually make it take longer for many people (and, I would predict, especially for the Ennea-type that tends to be attracted to Less Wrong).

But in my entirely non-expert opinion, I think pretty much all of developing empathy comes down to: (1) recognizing that what others do is always sensible to them and (2) being sincerely interested in the way in which what they're doing is sensible to them. I would just add that it's really important that this be done in a non-judging way. It's fine to decide that others are daft and reject the reasoning they use to see some behavior or belief or what-not as sensible, but that doesn't help you at all to understand the world as they experience it. Really understanding others' perspectives requires being open to seeing the world that way, even if you later reject that vision when no longer donning it.

Disclaimer: The above is based on my personal experiences plus a general impression I've gotten from various research articles I've read. The fact that this has been working nicely for me has put my "seek out biases in this area" to-do pretty low on my priority list. I could be completely off my rocker here. If so, please let me know!

Some hallucinogenic drugs (and perhaps other drugs) can make empathy a lot easier. How to take away skill development for use during sobriety, I'm not sure - I haven't experimented enough to have any insight. But I suspect it's possible. Relevant LW post

It seems to be almost universally held that empathy is a desirable personality trait. I can certainly see that having better theory of mind - being better able to predict other peoples actions - is useful in any situation. But empathy, to me at least, also has connotations of sympathizing with the other person. Whilst I can see that this would be very useful in certain situations (e.g. sexual relationships), it seems to also be potentially harmful in other situations (e.g. management.) For example, firing someone who has been a reliable worker for many years, or confronting a person whose work is sub-par are all things that could be made more difficult by being sympathetic. Whilst I'm sure its possible to overcome such a reluctance consciously, these feelings might cause you to shy away from even thinking about such things, which cannot so easily be consciously corrected.

I'm also curious about how empathy brings benefits in social interactions apart from the theory of mind aspect of it. Obviously, people will like people more who seem to agree with them, like them and understand them; but it seems that all these signals could, with moderate effort, be faked effectively. I wouldn't consider this to necessarily be deceptive, either -- the signal one is sending is communicating how much one likes the other person and wishes to be allied with them; provided you genuinely do like the person and wish to be friends with them, then the signal seems to me to be honest. But its possible that I'm missing some subtle aspects of empathy that cannot be so easily faked.