Imagine a box of balls. They have size, weight, colour, density… etc. These properties, each a measure, are dimensions and they are orthogonal to each other. Taken together are multi-dimensional.

Now take a set of words. They have "sizeness", "weightness", "colorness" and "densityness"...and "pythonness" and "haskellness" and even "adjectiveness" and "verbness" and so on and so forth...You can readily see that this can encode arbitrary patters (like language grammer or program syntax)

Training identifies these dimensions in the training data and links it with each word/token. Then given a stream of such tokens, each with its own set of dimensions (which can be huge), and LLM predicts the dimensions that the next token is most likely to have...