This is an excellent question!
The construction you are using is the accusative and infinitive. This is an English-language construction that is derived from the Latin accusativus cum infinitivo. [1]
Here are some examples of the accusative and infinitive.
- The principal asked them to dress appropriately.
- Ask them to dress appropriately.
You can see that the verb is in the infinitive form ("to dress") and that the subject of the verb is in the accusative case ("them").
There is a class of verbs that can use the infinitive without the particle "to". This is often called the "bare infinitive", and it is contrasted with the "full infinitive" or the "to-infinitive".
The verbs let, make, do, help, can, will belong to this class of verbs. They can all use the bare infinitive.
- Let them begin!
- Help them finish their homework.
- Tell me a joke and make me laugh.
The phenomenon you are referring to is the omission of the particle "to" in an infinitive phrase. This can happen when the infinitive is preceded by specific verbs like let, make, do, help, can, will.
I think this happens for euphonic reasons.
Consider the sentence "I listen to him lecture the class." It would be awkward to say "I listen to him to lecture the class." So that is one example where we omit the particle "to" for euphonic reasons.
Consider the sentence "Let them begin!" This can be said very quickly. If we added the particle to, as in, "Let them to begin!" then it would be difficult to say.
One important consideration in language is how easy it is to say a word or sentence. We sometimes omit the particle "to" from an infinitive phrase when it is easier to say it that way.
Footnotes:
- The accusative and infinitive is often abbreviated as ACI (due to its Latin antecedent) or A+I.