Think of walking in a field with tall bushes which have thorns. Like blackberry bushes. The thorns will attach to your clothes. So snag is when the thorns grab onto your clothes.

The word can also be used in a romantic context, as in "she snagged herself a great guy".

Aside from what the others have said, I think you might also say, for example: "A coworker snagged me into another problem before I could leave the office." Using "to get" makes it a passive form. "I got snagged into another problem at work before I could leave." It leaves the "who" unstated.

"Snagged" is another word for being caught or taking possession of something or someone, usually in the physical sense. Examples: I snagged a new pair of sneakers during that sale. He snagged him(self) a new girlfriend. My bookbag snagged on the something sticking out. Also, you can reference Britannica.com and search "snag".