One of the things that always impressed me about the RTD run of DW (and the classic series) was its ability to introduce new characters and fully flesh them out, giving you a good sense of their histories, personalities, motivations, and dreams. Sometimes these characters are central to the story (Professor Lazarus in "The Lazarus Experiment", Luke Rattigan in "The Sontaran Stratagem"/"The Poison Sky"), sometimes, they provide a secondary narrative (the Caecilius family in "The Fires of Pompeii"), and many times, we only get a tiny glimpse of them as they flit in and out of the story (Suki in "The Long Game"). HN/FOB provides a whole host of these characters as we observe daily life in the Farringham School for Boys and how it gets destroyed by the arrival of aliens.
"Aliens", of course, includes the Doctor, and as Joan points out at the end, the other aliens would not have come there if he hadn't been there first. And yet, the story isn't about the Doctor; it's about a human named John Smith. It's about the wonders of normal human life and how one man deals with finding out that his life isn't what he thought it was. The episode builds up his life and his dreams, then dashes them on the rocks, and he's forced to choose to make the ultimate sacrifice when he doesn't even know if it's worth it.
Then the Doctor returns, and in contrast to John, he's completely alien, something we rarely get to see. He's callous and cruel, and I'm talking about how he treats Joan, because he's not human and does not understand her at all. The episode is well-crafted to remind us that he might seem to be human but he's not, and this might not always be good.
This is why this episode is my favorite: it's an exploration into both the human and the Time Lord psyche, expertly woven into an engaging story filled with interesting and disparate characters.
There are a number of other individual aspects of the episode that also appeal to me. First, it continues the thread during series 3 of the Doctor taking Martha for granted and asking of her more than he really should. Within that idea, it doesn't ignore the realities of her situation. She's discriminated against for both her station and her race, and it's treated as facts of everyday life, which it was back then, and this, of course, makes the whole situation much harder for her. Second, this episode gives us a tiny glimpse of the power of a Time Lord. The Doctor rarely ever does anything a normal human can't do, but we know that the Time Lords are powerful, either through their own abilities or through their technology. Without showing it directly, the episode tells us that the Doctor subdued the Family of Blood singlehandedly, and then proceeded to mete out extraordinary punishments. This power that the Doctor keeps hidden away adds to his mystery; it feels very much like a nod to the Cartmel masterplan, actually. (Which only makes sense, since the original novel by Paul Cornell on which this story is based comes from it.) And lastly, there's that cruelty, that pull towards the darkness that this incarnation of the Doctor exhibited from the very beginning and eventually succumbed to. That was just beautiful to see.