This is so seriously unadvertised it’s almost criminal (and is it, under the ADA? I don’t know.).
Anyway, for titles on Netflix that support it, with Apple TV’s newest 4.4.2 update (I don’t know if it was supported earlier than that – I just figured it out), you start playing the title, then you press the up key and then hold down the select key (for more options). This should take you to a menu where you can select subtitles, if any are available.
I started looking for this after I noticed, after my iPad’s iOS 5 upgrade, that the Netflix app on that platform was also finally supporting subtitles.
This isn’t so much an issue for me (though I like them in general – though I sometimes feel like they give away punchlines) but for my Dad it really does matter a lot.
UPDATE: Also after the title starts playing, if you just click and hold down the select key (the one in the center of the directional button ring), the subtitles screen should immediately show up (if any are available for your title).