Here's a quick look at the four quotations you ask about.
When someone asked Berra how he liked school, he said, "Closed."
Berra was not a scholarly guy, and he wasn't a huge fan of school. But when someone asked him how he liked school, instead of saying "Not much" or something similarly focused on what his feelings about school were, Berra interpreted the question as asking him to identify the conditions under which he liked school most. "Closed" means simply "I liked school most when it was closed." The misunderstanding of this question to comic effect is reminiscent of an anecdote in which someone asked Diogenes, the famously penniless cynic philosopher/gadfly of Athens (and other parts of Ancient Greece), what his favorite wine was. His answer: "Other people's."
Take it with a grin of salt
This is simply a malapropism in which Berra either misstated or misunderstood the idiomatic phrase "take it with a grain of salt" (meaning don't accept it at face value) as containing the word grin instead of grain. The fictional Mrs. Malaprop in Richard Sheridan's play The Rivals repeatedly commits similar errors, such as referring to a chivalrous person as "the pine-apple [instead of pinnacle] of politeness."
It gets late early out here
This a logical paradox because late and early are opposites. The idea that one happens on a schedule defined by its opposite is difficult to make sense of unless you change the terms of the statement. There are at least two ways to apply a meaningful interpretation to what Berra said. One is to imagine that he was playing a game in a city farther north than where he was used to playing, and that sundown came earlier than he expected; we could rephrase his comment more accurately in that case as, "It gets dark early up here." Another possibility is that he was visiting a city where the nightlife (bars, restaurants, clubs, etc.) shut down earlier than it did in New York; we could capture the gist of his comment in that case as "Businesses close for the night early around here."
I think they just got through marinating the greens
This is another malapropism. In all likelihood Berra was trying to come up with the word irrigating or the word manicuring. The joke is that marinating, though it falls roughly halfway between those two words, has nothing to do with golf or with golf greens.