Would not be able to versus will not be able to

Because the sentence is in the past tense. From the point of view of the past tense, the future is expressed with "would" rather than "will."

If the sentence began with "Linda phones and explains that," you'd use "will not be able."

In a comment, John Lawler wrote:

Because it's slightly more polite in giving an excuse. 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.