LLMs learn from examples where the logits are not probabilities, but how a given sentence continues (only one token is set to 1). So they don't learn probabilities, they learn how to continue the sentence with a given token. We apply softmax at the logits for mathematical reasons, and it is natural/simpler to think in terms of probabilities, but that's not what happens, nor the neural networks they are composed of is just able to approximate probabilistic functions. This "next token" probability is the source of a lot misunderstanding. It's much better to imagine the logits as "To continue my reply I could say this word, more than the others, or maybe that one, a bit less, ..." and so forth. Now there are evidences, too, that in the activations producing a given token the LLM already has an idea about how most of the sentence is going to continue.

Of course, as they learn, early in the training, the first functions they will model, to lower the error, will start being the probabilities of the next tokens, since this is the simplest function that works for the loss reduction. Then gradients agree in other directions, and the function that the LLM eventually learn is no longer related to probabilities, but to the meaning of the sentence and what it makes sense to say next.

It's not be chance that often the logits have a huge signal in just two or three tokens, even if the sentence, probabilistically speaking, could continue in much more potential ways.

There are some that would describe LLMs as next word predictors, akin to having a bag of magnetic words, where you put your hand in, rummage around, and just pick a next word and put it on the fridge and eventually form sentences. It's "just" predicting the next word, so as an analogy as to how they work, that seems reasonable. The thing is, when that bag consists of a dozen bags-in-bags, like Russian nesting dolls, and the "bag" has a hundred million words in it, the analogy stops being a useful description. It's like describing humans as multicellular organisms. It's an accurate description of what a human is, but somewhere between a simple hydra with 100,000 cells and a human with 3 trillion cells, intelligence arises. Describing humans as merely multicellular organisms and using hydra as your point of reference isn't going to get you very far.