One of the most important things that playing in a HTHD game requires of everyone involved -- players and GM -- is commitment. And when I say commitment, I mean a willingness to "go deep" into emotional territory that most roleplaying games don't require or endorse.
As an aside, it also requires commitment in more pedestrian ways. You will find, in a game that focuses on character development and drama, that it's increasingly difficult to go ahead with a session if one or more players is unavailable. The characters are intimately tangled up in every aspect of the story, and more than that, the players are intensely involved on every level of the story -- although you could have a session where you do a "side story" focusing on some of the characters, players are also committed to the game on the level of audience. Missing an important part of the story is a problem, and you can't just catch up by downloading the episode from the internet.
Most players are initially reluctant to open themselves up to the level of commitment that HTHD demands. The games they are probably accustomed to tend to avoid situations of emotion and drama, and the idea of allowing their character (and, by extension, themselves) to become vulnerable in a roleplaying game is strange and scary. And it's true that even for players with experience at this sort of thing, often your first instinct is to back away from an in-game conflict rather than tackling it head-on.
Sometimes this is an instinct that a conflict shouldn't be resolved / grappled with at that point in the story, or a genuine uncertainty about what to do / how to approach the conflict, but most of the time it's just a fear of commitment to the scene. Drama thrives on uncertainty and unexpected changes in personal dynamics; it's always a good time to take on a conflict, because the fallout from a scene should give you material to move forward and develop things further.
Megan summed up the technique that you need to get past that initial moment of pulling back from a dramatic moment with an anecdote from her theatre days. Whenever you feel yourself starting to pull back from a scene, you say "Fuck it!" and go for it.
The FUCK IT! School of Acting.
Often, simply being aware of this instinct to pull back from a scene puts you in a position to ignore that instinct. Just remember that salty mantra and dive right in the next time you find yourself avoiding something that could be interesting and exciting in a scene.
Commitment is always the more interesting choice. Powering through those awkward moments and seeing where they take you is the stuff of good drama and good gaming. Games like Primetime Adventures make this a requirement of play by building into your games a structured character arc with defined "spotlight" episodes for each character, but NOW is always a good time for drama.
Remember: Say "Fuck it!" and go for it.
Excellent advice for players, and useful stuff for GMs too.