If you were to just pick up a new color and start knitting, you would eventually get a big gap between the two colors in the knitted piece, as you are then essentially knitting two separate pieces. (Go ahead. Try it. I shall wait.)

See? To prevent that, when doing intarsia on a vertical join, twist the yarns of the two colors together on the wrong side of the fabric. You will twist the yarn every time you do a color change, but all the twisting is done on the wrong side. Here is a little photo tutorial on how to do that:

I have shown the wrong side of the fabric to make it easier to see, but you twist the yarns the same way if you are knitting on the wrong side.

I have knitted the four buttonband stitches in dark blue and now want to pick up and knit the aqua stripe.