Justice Thomas is essentially arguing that the 14th Amendment (the "no depriving any person of life, liberty, or property, without due process of law" one) is the wrong reason to give the person a refund of their fees after an overturned conviction.

Reading the dissent, it basically says: when the person was convicted, they paid their fine, and at that point the fine money is now property of the state. The overturning of the original conviction doesn't, however, mean that the fine money also reverts ownership - he claims it is still in ownership by the state so the person cannot claim it is their property being deprived without due process under the 14th amendment.

He proposes that the "correct" way to get the money back is by invoking this other thing called "The Exoneration Act," NOT the 14th. The problem is the Exoneration Act isn't a law that says the state HAS to return funds, it's more of a specific appeal process for people to apply to get funds back. In other words, he recognizes there is no state law in Colorado that requires refunding these fees (nor does he offer a viable Federal one), but he claims there is a process available, at least.

In that sense, his dissent isn't really about him believing these people are NOT owed a refund in the moral sense of what is just, he's simply arguing that the court should not establish that refunds of costs/fines in cases where convictions are overturned is guaranteed by the 14th Amendment. It's kind of a pedantic dissent. And I still don't agree with him :)