ta med sig is a reflexive verb. It's like this:
jag tar med mig 'I am bringing'
du tar med dig 'you are bringing'
hon tar med sig 'she is bringing'
vi tar med oss 'we are bringing'
ni tar med er 'you are bringing'
de tar med sig 'they are bringing'
So dig is part of the verb here.
This is the word order my cousin Gerda always used in English. "Take this with you home, honey." Sweet lady. She left Sweden at age 15 alone, took a boat and then went by train 1000 miles to live with relatives in Chicago. I used to clean her house for her when she was elderly and I was a teenager, and read her Swedish mail to her during lunch. I had no idea what I was reading, but I knew how to pronounce it and she understood. Nice memories.
Actually, it was more like, "Take diss vitt you home." The accent of all of my grandparents and great aunts and uncles, and this distant cousin.