Looking at Cambridge Dictionary that JavaLatte mentioned, we can answer your question about where advers like angrily (manner) should usually go:
They usually go in end position.
They sometimes go in mid position if the adverb is not the most important part of the clause or if the object is very long.
The end position of the clause is the last item in the clause:
Why do you always have to eat so fast?
The mid position is between the subject and the main verb:
Apples always taste best when you pick them straight off the tree.
Where there is more than one verb, mid position means after the first auxiliary verb or after a modal verb:
The government has occasionally been forced to change its mind. (after the first auxiliary verb)
You can definitely never predict what will happen. (after a modal verb)
We mightn’t ever have met. (after the modal verb and before the auxiliary verb)
Following this rule and considering how long the object of the sentence is I'd say it should go in the middle position between the subject the student leader and the main verb explained:
E. The student leader angrily explained why he had to leave the panel discussion forum.
Which isn't any of the available options.
There is still another possibility. In the previous example the angrily modifies the way the student leader explained something. It's also possible that the angrily should modify the way he left the forum. In this case we need to look at the subclause ** why he had to leave the panel discussion forum** and see where the adverb should go.
Here the object to leave the panel discussion forum is still pretty long so the adverb should go in the middle again. Given how we have two verbs have to and leave it should go in between:
F. The student leader explained why he had to angrily leave the panel discussion forum.
Which looks really close to answer D except that the to belongs with had here, not leave. We can see this by replacing have with should for example:
The student leader explained why he should angrily leave the panel discussion forum.
One last thing to point out is that the rule to put the adverb in the middle could also be taken as meaning the adverb isn't the most important part. So if we really want to emphasise the action being performed angrily we should put it in the end instead:
B. The student leader explained why he had to leave the panel discussion forum angrily.
Notice tho, that in this case the angrily modifies leaving the forum. The student leader could be explaining this afterwards in a calmly manner. For example if he wanted to make a statement there and for people to take that statement seriously he had to leave angrily.
The student leader calmly explained why he had to leave the panel discussion forum angrily.
If instead we want to enforce the meaning of explaining it angrily and it really is important that it is angrily we could go for option C instead:
C. The student leader explained angrily why he had to leave the panel discussion forum.
Without further context I see no reason why B shouldn't be the right answer tho.