Heavy.com tried to dig that for you:
During the scene, it looked like Bill was burning a pair of jeans, a shirt, a jacket, and a blue flannel top.
The flannel top is especially obvious in the scene where it’s burning.
And it’s this flannel top that holds the clue about what Bill’s burning. Some fans think it’s Monty’s top since he was wearing a flannel top on the night that Bryce died. But it’s actually more likely to be Alex’s shirt. Check out what Alex was wearing the night that Bryce died.
Also when he said he wished it was his head, not Alex:
I wished that bullet was in my head, not yours. You understand?
which shows he can do anything to protect him even a bullet to his head. Also, he is in the police department so we can guess he is good with paying attention to details, especially on a big day.
Also, sometimes parents are like that, my mom used to remember what I wore weeks back when even I forget sometimes.
But now your major question, why he burned it?
Most probably to protect Alex and indicating he knows the truth. We can't be sure that his cloth had no blood on them, Bruce was wounded when he was throwing him and Standall being a cop can't take chances.