I sure hope this isn't confusing, because I think it could be a little more concise. Here's why you need to join a second ball of yarn:
Let's say the 103sts are the first 4 X's, and in the center there, is the 31 sts you've bound off, and then the remaining X's represent the sts left (knit to the end of the row).
The second ball of yarn is attached to work on the side you've now made lone, due to the bind off in the center. (You can't bring the yarn across a neckline to knit it, otherwise you'd have yarn going across it).
By working both sides at once you're going across one segment, ignoring the void created by the bound off sts, and then doing the other side. So technically you're not knitting them at the same time (as if it's a continuous row), but in order. If you didn't work them both at the same time, you'd get this:
I really hope that makes sense!