There are some people who shudder at the mention of grafting. Not too long ago, I was one of you, dear friends. But, between a couple top-down socks and a tubular bindoff or two, a lightbulb came on. Something clicked. And now I know it, and I understand it, and I even appreciate Kitchener stitch.
It used to be that every time I wanted to graft an armpit, sew up a toe, or any of the other numerous uses for kitchener stitch, I had to look up the step-by-step instructions on the web and follow them slavishly. It was painful. I was already uncomfortable, having the sewing needle in my hand, having cut my yarn, worrying whether I cut it long enough, just wanting to have my darn socks done already...
Let it go. Leave all that anxiety behind. The trick is simple: you just have to understand it. Once you understand, you will never need to look up instructions again.
Here is my trick: You're going through each stitch twice. You set each stitch up by working them opposite (that is, purlwise for knits, knitwise for purls). The stitches come off of the needle when you are working them as they're supposed to be: that is, the (front) knit stitches knitwise, and the (back) purl stitches purlwise.
Remember the manta Knit off, purl on. Purl off, knit on. For your front set of stitches, you are taking off the knit stitch by going through it knitwise, and setting up the stitch behind it by going through it purlwise. Then you move to the back stitches, and take the first off purlwise, and set the next stitch up knitwise. And then you repeat.
I think one of the most confusing things for me was that, in most sets of instructions, there is a "set up" before you repeat the "knit off purl on purl off knit on" instructions. Now that I can understand it, there's no problem for me: you have to go through each stitch twice, so it makes perfect sense.
I have memorized Kitchener!!! And, hopefully, now you can too.