Got rear ended

[jaedub]

Sad sad night. I was making a right turn into a Chipotle driveway when all of a sudden a (2003 Honda Accord) lady struck me from behind. She thought I was going straight and fail to notice that I was slowing down to make my turn.

Accident (2).JPG
The impact wasn't exactly a straight on collision. Because I was turning, the hit was angled which is why there is a major scrape.

After exchanging our papers, she called her brother and they both begged me not to go through the insurance. They said they'll cut me a check after I get a quote. The way they were begging seemed a bit suspicious for some reason.

My question is:Should I go through the insurance or have them pay out of their own pocket?

Am I in any risk?

This is my first encounter of an accident so any help/tips are appreciated.