Before Finn I used to sing alone.
That jacket is a talisman that allows you to grieve.
Carole refuses it. She shakes her head and puts it in the donation box because she is not ready for that level of release.
Kurt puts it on and has his moment of grief, in the midst of a week where he’s everyone else’s guide and center and rock.
Santana blocks out her grief, turns it to anger, and lashes out at everyone. But when Kurt places the jacket on her shoulders, she finally is able to let her true feelings come out.
And similarly with Will. He’s been hiding from what he feels because he thinks he needs to be there for the kids. And it’s that jacket, furtively stolen, that lets him finally grieve.