How does the word "to" function with an infinitive?

Solution 1:

In an infinitive clause, the function of the infinitive marker to is to mark the infinitive. Seriously.
Specifically, it marks (introduces and identifies) the Verb Phrase of an infinitive clause, which always starts with the infinitive form of some verb (either the main verb or an auxiliary verb).

To is part of the for-to, or "Infinitive" Complementizer; for marks the Subject, to marks the VP.
However, most infinitives do not have overt subjects (though it's always clear what the subject is, even if it's only Indef), and when the subject is deleted, so is its marker for.

• It's time for us to leave ~ It's time to leave ~ *It's time for to leave

The for is only necessary at the beginning of a sentence, with an infinitive subject clause:

• For him to leave soon would be a good idea.
• *Him to leave soon would be a good idea.

As can be seen, omitting the for above fails to mark the infinitive clause as an infinitive clause.
Similarly, the to is required to mark an infinitive verb phrase as infinitive. Why is this necessary?
Because English has lost its inflections. Every other European language that uses infinitives has a special inflection that marks them. English doesn't -- the English infinitive (with the exception of be, which is almost always irregular) has exactly the same form as the present tense (with the exception of the third person singular present -s): to go/I go; to sit/we sit; to move/they move;
but: to be/I am/he is/we are; to go/he goes; to sit/she sits; to move/it moves.

So to essentially takes the place of the Germanic -en/-an infinitive suffix that was lost in English.
When a language loses morphology, syntax comes in to take over. That's what prepositions and articles and complementizers and auxiliary verbs in English are all about; they're filling in the gaps where other languages use inflections.

The upshot is that infinitive clauses often lose subjects, so the norm is for the for to be missing; thus its presence is exceptional. On the other hand, infinitive clauses always have verb phrases, so the norm is for the to not to be missing; and its absence is exceptional.

That, in fact, is the ultimate source of the "split infinitive" zombie rule. It's so obvious that the English infinitive complementizer to has the same function as the Latin infinitive suffix -re that 18th-century grammar mavens decided they were spiritually identical or something, and therefore, since you couldn't split the Latin infinitive, you couldn't split the English one, either.

And, as usual, everyone believed this. Because grammar.