What is the difference between:
Someone care to explain what those things are?
Someone care to explain what are those things?
Are these both correct? Which one should I use?
The formation of a noun sentence using what is superfluous. The use of what as question word is not needed either, because the grammatical structure already marks the question.
Someone care to explain those things?
Your second form is wrong, because it needs a colon or at least a comma, because what introduces a main clause that can stand on its own. But then, two main clauses in one sentences is just bad style if they have the same meaning. The first form is a nominalization.
Take "the decision what question to make the main clause". Expanding the coloquially shortend phrase to a more formal one, we have "the decision about which should be the main clause" vs "the decision about which the main clause should be". The latter would clearly sound too close to the decision, which the main clause should be, although the main clause can't actually make any decisions.