The lesson is about using the Future Anterior in the clauses with quand/lorsque/une fois que/and aprés que when those clauses are used in a  multi clause sentence denoting FUTURE actions  and ONE  ACTION must be done(executed) before the others start.  The one which must be done FIRST(and so will be anterior to the others) is in the Future Anterior tense. This is formed using the conjugated future tense of ETRE or AVOIR and the past participle of the relevant verb.

 ETRE (serai/seras/sera/serons/serez/seront)  + particple.

 AVOIR (aurai/auras/aura/aurons/aurez/auront) + participle.

This should explain what you refer to as the verb 'aura'

Avoir and Etre have irregular stems so their future tense looks funny but luckily the endings look exactly like all verbs in the future tense... and also note in the same sentences you WILL see the same future tense endings because the other clauses in the sentence will be using that tense.

I find grammar explanations less helpful than examples. Whats going on here is called the Future Anterior because the first part of the phrase eg in I will have done is the the future of to have - I will have, j'aurai. The second part is the past participle of to do, faire, done, fait. Hence I will have done. J'aurai fait.