If something is snagged, it is caught on something else, like a piece of clothing getting caught on a nail.
Snagged can make people think of getting stuck or having a problem.
Examples:
Her sweater got snagged on the fence.
I snagged my pants on a branch.
The kite snagged on the tree and wouldn't come down.
I snagged my sock on the chair and fell over!
The balloon snagged on the roof and wouldn't come down.
He snagged his jacket on the doorknob and couldn't get it off.