Close your eyes for a second. Try to find the thinker. The one having the thoughts. The "I" that's about to think the next thing. Look hard. What you find is more thoughts. The looking is itself a thought. The wondering is a thought. The "I'm trying to find me" is a thought. There's no thinker separate from the thinking. There's just thinking with an "I" attached to it.
K's claim is that the thinker is an invention of the thought. The thought needs a host to make sense of itself, so it produces one. The "me" that says "I am thinking" is itself a thought. The illusion is that there's a stable thinker who has thoughts as possessions. There's no possessor. There's only the possession, claiming to have an owner.
Once you see this, the structure of "I" shifts. You stop being a person who has thoughts. You become the place thoughts happen. There's a big difference. The first is a defended center with stuff on the table. The second is a clearing. Thoughts arrive. Thoughts pass through. Thoughts depart. None of them are "me." The "me" was a label thought put on itself to keep itself going.
Some people find this terrifying. The fear is the thought-of-self defending its existence. Others find it instantly familiar — like a thing they suspected but had no language for. The body usually relaxes when the move lands. There's less to defend.
There is no thinker behind the thoughts. The thoughts are the thinking. The "I" is one of them, walking around acting like it owns the place.