His buy out was negotiated.
His buyout was written into his contract, like all buyouts for termination without cause are.
For Muschamp, the buyout for termination without cause is $2M/year for the remaining three years of his contract plus the prorated amount for the rest of this month and the month of January 2018. That's what got him to a little over $6.3M, paid over 5 installments, the first of which comes next month and is 50% of his buyout. The remaining 4 installments are 12.5% paid annually on the effective date of his firing. No mitigation clause exists anywhere in the contract. I have no idea how there are still college programs agreeing to contracts without a mitigation clause in it.
Compare that with Bo Pelini's contract, which not only has a mitigation clause but requires Pelini to mitigate the damages. He has to find a job in order to continue receiving his buyout, which is $150K/month for the duration of his contract, less the base monthly salary at his next job. If Bo Pelini had gotten the same salary Will Muschamp is getting at Auburn, Nebraska would barely owe him anything.
Will Muschamp has an amazing agent to get a contract that coach friendly. Better than Charlie Weis' agent and that's saying something.