The “inc in next st” means for you to -- in one stitch -- knit the stitch but don’t remove from needle, then yarn over, knit the stitch again and then remove from needle. This will result in three knit stitches in a row becoming five in the pattern. Eventually, it’ll give you seven knit stitches in a row. And you’ll decrease that a couple of rows later with k3tog stitches.

(Taken from Roachelly’s comments on this pattern. Thank you Roachelly for clarifying this.)