Teddy introduces this concept to Nicholson towards the end of their conversation on the sun deck. When he tries to talk to Nicholson about "getting out of the finite dimensions," Nicholson responds as most of us would: a block of wood has clearly defined length and width, an arm is just an arm, and so forth. Teddy's response is that Nicholson is only using logic – something we should dispose of if our goal is spiritual transcendence. "Logic's the first thing you have to get rid of," says Teddy (4.91).
To understand this better, you should check out Shmoop's "What's Up With the Epigraph?". There, we explain the difference between understanding something logically and understanding something spiritually. 2+2=4 is an axiom you can understand logically. Logical and mathematical truths are true no matter what – they are said to be true a priori. However, the Zen riddle "What is the sound of one hand clapping?" is something you have to understand spiritually. There is an "answer" to this sort of question, but not one that you could explain in words to somebody else.
Teddy is getting at this very distinction. The important things, he explains, aren't things we can understand the way we understand 2+2=4. The important things are questions like "What is the sound of one hand clapping?"
Not only is logic unimportant, but it actually gets in the way of spiritual understanding. Teddy thinks of logic as a limitation on mankind. This is where the famous apple of logic comes in:
"You know that apple Adam ate in the Garden of Eden, referred to in the Bible?" he asked. "You know what was in that apple? Logic. Logic and intellectual stuff. That was all that was in it. So – this is my point – what you have to do is vomit it up if you want to see things as they really are. I mean if you vomit it up, then you won't have any more trouble with blocks of wood and stuff. You won't see everything stopping off all the time. And you'll know what your arm really is, if you're interested." (4.99)
It can be a little confusing to understand what Teddy means about things "stopping off all the time." To really unpack this in detail, you'd have to do some research into Eastern religious philosophy and start with a solid grasp of its basic tenets.
One surface way to approach this is to consider Teddy's claim that "everything [is] God" (4. 75). When he saw his sister drinking milk, he realized that "she was God and the milk was God" and that "all she was doing was pouring God into God" (4.75). This is an example of "getting out of the finite dimensions." When you look at a little girl drinking milk, you see a clearly defined glass of milk and a very separate little girl. These are two distinct objects – there is a line where the girl ends and the glass of milk starts. What Teddy is saying, however, is that this is not really the case. Because you're looking at the scene logically, you see finite dimensions. But if you try to understand the scene spiritually, you don't see these clearly defined boundaries.
Teddy again mentions the apple in his discussion of education. Before poisoning children with the apple of logic, he says, you ought to let them discover the world with their own eyes, free of logic and intellectual boundaries:
"I wouldn't start with the things schools usually start with. […] I'd first just assemble all the children together and show them how to meditate. I'd try to show them how to find out who they are, not just what their names are and things like that . . . I guess, even before that, I'd get them to empty out everything their parents and everybody ever told them. […] I wouldn't even tell them an elephant has a trunk. […] I'd let them just walk up to the elephant not knowing anything […] about it. […] I wouldn't even tell them grass is green. Colors are only names. I mean if you tell them the grass is green, it makes them start expecting the grass to look a certain way – your way – instead of some other way that may be just as good, and may be much better. […] I'd just make them vomit up every bit of the apple their parents and everybody made them take a bite out of." (5.10)
(A fun aside: Interestingly, two characters in Salinger's novel Franny and Zooey tried just this educational approach with their younger siblings – with a somewhat problematic consequence. Guess you'll have to read the novel to find out more. If it helps, know that one of those characters – Seymour Glass – might be the artistic source for the character of Teddy.)
Back to the matter at hand: though Teddy seems to have vomited up quite a bit of the apple in question, he explains that most people in the world have not. "The trouble is," Teddy says, "most people don't want to see things the way they are. […] I never saw such a bunch of apple-eaters" (4.101). There are quite a few apple-eater examples in the story – Booper, Mr. and Mrs. McArdle, and even Nicholson (see his "Character Analysis") – but Salinger may be making an example of his apple-eater readers, as well.