Why is $BPP$ closed under complement?
It follows directly from the definition of BPP.

The definition says that a problem with a yes-or-no answer is in BPP if there is a randomized algorithm that gets the correct answer with probability $\ge 2/3$ on all possible instances.  This means that on all instances where the correct answer is YES, the algorithm has to output YES with probability at least $2/3$; and on all instances where the correct answer is NO, the algorithm has to output NO with probability at least $2/3$.

Therefore, if a problem $P$ is in BPP, its negation is also in BPP.  Take the algorithm for $P$, and just flip its output.  That will be a valid algorithm for the negation of the problem.