Being somewhat new here, I don't think I'll know all of the etiquette required of me here, but I'll have a stab at presenting my opinions (and whatever limited knowledge I might have) on the BNP.
It's well established now what they stand for (white Britain, as if a 97% racial majority isn't enough) and I don't support them at all, their policies and arguments hold little water for me. However, it's been observed repeatedly that in times of hardship, people lean more towards radical ideals and actions (the rise of the Nazis, Soviet Russia and perhaps even Napoleons France are examples of this). As we are witnessing recession (and in Britain it stands to be worse than others, Germany for example), combined with the free publicity the established parties and the media so willingly give them, along with a growing culture of fear and simmering discontentment with the governing body will help push their agenda somewhat.
They say what they think people will want to hear (as they should if they want to succeed), and there is genuine concern over immigration, the EU and so on. With a population now over 61 million (which the Daily Mail so blithely attributed to immigrants), it is perceived that we, as an island nation, cannot sustain much more (without imports, I sincerely doubt we could sustain much more than half of that, although I am uncertain of true numbers).
A lot of people (myself included) see the EU as an interfering, over-regulating entity that spares little thought for the average citizen and seems to do more harm than good (though I do admit, with a weakening pound and a very limited economy, integration may well be inevitable).
The other side of the coin, as has been said before, is the other parties, namely Labour, Conservatives et al. The UAF, seemingly ignorant of the fact that it combats fascism with, well, fascist techniques (suppressing freedom of speech and movement), draws unnecesary attention to the BNP. Just ignoring them, whilst not treating the populace as idiots, will weaken the BNP and thus reduce the threat they pose. This, and that the big 2 seem hell bent on bringing this nation to its knees and willfully ignore the aggrieved voices they're supposed to govern. Such tactics only fuel the fires and so the fringe parties grow stronger.
And that leaves me in a pickle really. I'd never vote Labour or BNP and whilst I have more conservative leanings, I wouldn't trust the Tories as far as I could throw them. Lib Dems are also rather uselss. I believe this is a situation I'm not alone in either.
Tl;dr? Ok, BNP become prominent because of hardship, ignorance, displeasure with the current government and because they seem to listen. That's what I think at least anyway.
Incidentally, I'm not buying a complete comparison with the Nazis. Granted, the racial view is the same, but the Nazi's had some pretty sound economic fixing that helped Germany pull through the Depression and a heavy war debt from the previous conflict. I doubt the BNP, in its current form, have such a card to play.