I am also interested in the explanation. I have a guess but would like to have it confirmed: burada -> sit as in "why have you been sitting here", buraya -> sit as in "why did you seat yourself here?"
If that is correct, I think the (one) correct translation of "Neden buraya oturdunuz?" could be "Why did you seat yourself/yourselves here?"
I think in the meantime I have learned enough Turkish to answer my own question :) The nuance between accusative & dative case is:
1) Neden burada oturdunuz -> Why were you sitting here?
2) Neden buraya oturdunuz -> Why did you sit (down) here?
I did not translate the tense correctly, in order to make the difference in meaning clear. The correct translation in both cases is "Why did you sit here?"
@blainluckyk, burada has the locative ending -da, where buraya has the dative ending -(y)a. It's a distinction like German hier/her, Esperanto ĉi tie/ĉi tien, archaic English here/hither, etc. The confusing part is why this sentence uses the dative; it doesn't show movement toward something but a location.