How does the Modern Greek pronoun το modify verbs?

As a pronoun, το is the clitic accusative neuter third person pronoun, and it corresponds to “that” or “it”. So, ξέρω “I know”; το ξέρω “I know that”.

Which means that, in the first instance, το is not modifying the meaning of a verb; it is completing it by providing an explicit object.

You could argue that in this context, το is pretty vague, and it is. Moreover, “know” is a transitive verb—you always know something.

The distinction here is subtle. “I know that” is explicitly referring back to something just said. By not giving an object, “I know” implies a more generic statement (and therefore is somewhat more defensive): “I know stuff (including that)”, “I am clueful in general”.

On the other hand, if you’re answering a question, such as “what time is it?”, you would answer δεν ξέρω “I don’t know”, just as in English, and not δεν το ξέρω “I don’t know that”. The pronoun does not refer readily to indirect questions. You would answer δεν το ξέρω referring to a specific entity; e.g. ξέρεις το τραγούδι που λέει ο Αστερίξ; “Do you know the song Asterix sings?”