I believe your pattern wants you to increase the number of stitches by casting on more stitches at the beginning of each wrong side row. If they wanted you to simply increase into next stitches, most patterns will say something like, "increase 1 in next three stitches, or something similar.
So, with your wrong side facing, do a knit cast on or cable cast on of 3 stitches at the beginning of the row, and continue purling the existing stitches to the end. Then knit one across without doing any cast on. All of your cast on stitches will be at the beginning of the wrong side (purl side) because if you look down to the RIGHT SIDE instructions, it says "Cast on sts at lower edge at beg of RS rows instead of WS rows."
So, at the beginning of the next 5 wrong side rows, you cast on 2 stitches instead of three.
After that, you are increasing only one stitch at the beginning of each wrong side row for 12 wrong side rows.
Here is a video for cable cast on: