Will not be able to simply refers to the future, without any presupposition. Would not be able to refers to the past tense of phoned and explained, so it's a standard indirect speech tactic.

Would not be or will not be?

"Won't" is the short form of "will not". 'Wouldn't" is the short form of "would not" and would is the past form of will. Won't and wouldn't are very common and informal in use, whereas will not and would not are usually formal.

Will be able VS would be able?

Saying "I will be able to help you," implies that you will help the person you are speaking to, and there is no obstacle in the way of you doing so. Saying "I would be able to help you," implies that you theoretically could help the person out, but there may be something that prevents you from doing so at some point.