Knitting dog sweater help!

I am quite new to knitting and working on a sweater for my dog. The pattern says to knit using a circular needle and follow a pattern. I am stuck when it comes to this part:
On the last round, 12 stitches before round marker start knitting with a new needle. Continue with new needle until you have 12 stitches after marker. (total of 24 stitches on new needle)
With original needle, cast off 4 stitches, work to last 4 stitches on original needle, cast off last 4 stitches
Work back and forth flat for 1 inch, remember to continue pattern with cables. Cut yarn.

I have put 24 stitches on a double pointed needle and now the thread is on the new needle. How do I go back to working on the original needle and what does it mean to go back and forth flat?

Thank you!

Welcome to KH!
What is the name of the pattern you are working? Is the cable pattern on the original needle or on the 24sts now on the new needle?