Welcome to Knitting Help!
It's the "K until 8sts on the needle" that's confusing. Usually it's written as "K 8sts" or "K until 8sts on right needle". Then that line would be:
Cast off 5 sts, K 8 sts
, turn, leave rem on a holder
(you would have 51sts on a holder but the yarn is coming off the 8sts which are now on the left
needle, WS facing, after the turn)
Cast off 3 sts, P to end.
Cast off rem 5 sts.
Rejoin yarn to rem sts, cast off center 38sts, knit to end, Complete to match first side, reversing shapings.
It's a very elegant pattern. Have fun finishing it and post a photo if you'd like. We'd love to see it.