Solution 1:

The particle "to" is not wrong in this sentence, but it is unnecessary. I would recommend against using it.

The phrase "to understand" can be interpreted as a special case of the infinitive; a kind of future infinitive or impersonal future tense. In that context, the first sentence means, essentially, "please help me develop an understanding of this (in the future)". While that may be technically correct, it adds nothing to the meaning of the sentence.

To add some weight to my argument, the COCA lists 142 entries for "help me understand" versus only 18 for "help me to understand". The results are similar for other constructions involving "help me ..." versus "help me to ...".

I think that the confusion stems from the way you must use the infinitive in other cases, for example: "I want to understand this", or "I am trying to understand this". In these cases, the particle is an absolute neccesity.

Solution 2:

Help is a special verb in that way - the to is usually dropped from an infinitive when it is modifying help. This form of infinitive is called the bare infinitive:

  • The bare infinitive is used as the main verb after the dummy auxiliary verb do, or most modal auxiliary verbs (such as will, can, or should). So, "I will/do/can/etc. see it."
  • Several common verbs of perception, including see, watch, hear, feel, and sense take a direct object and a bare infinitive, where the bare infinitive indicates an action taken by the main verb's direct object. So, "I saw/watched/heard/etc. it happen." (A similar meaning can be effected by using the present participle instead: "I saw/watched/heard/etc. it happening." The difference is that the former implies that the entirety of the event was perceived, while the latter implies that part of the progress of the event was perceived.)
  • Similarly with several common verbs of permission or causation, including make, bid, let, and have. So, "I made/bade/let/had him do it." (However, make takes a to-infinitive in the passive voice: "I was made to do it.")
  • After the had better expression. So, "You had better leave now."
  • With the verb help. So, "He helped them find it." (The use of the to-infinitive with the verb help is also common.)
  • With the word why. So, "Why reveal it?" (Use of the to-infinitive following why is also common.)
  • The bare infinitive is the dictionary form of a verb, and is generally the form of a verb that receives a definition; however, the definition itself generally uses a to-infinitive. So, "The word 'amble' means 'to walk slowly.'"
  • The bare infinitive form coincides with the present subjunctive form as well as the imperative form, but most grammarians do not consider uses of the present subjunctive or imperative to be uses of the bare infinitive.

Solution 3:

I think you may find there is also a difference here between US and GB English (a field worth writing a book about!). From experience, I'd say "help...to" is more prevalent in British English and less common in the US. As a British translator (from German) I've sometimes had "help to + infinitive" corrected to just "help + infinitive" by US editors.

Solution 4:

The particle to is what's called a Complementizer. It marks the verb following as an Infinitive (in English, that's necessary because English infinitive verb forms are identical with the present tense forms -- to go, I go; to sit, I sit, except for the single verb be (I am, to be). More on infinitive complements here

To is not a part of the verb that follows it, nor yet a part of whatever comes before it. It's a particle; one of those troublesome little words like the, that, of, at, etc. which English uses to decorate and distinguish its syntactic constructions, now that all its morphology's gone.

Infinitives get used in a variety of ways. One of them is as Complement clauses; these are Noun clauses that can function as the subject or direct object of a number of verbs. It's one of the principal ways we can form complex sentences. This particular chunk -- helping their son win the fight -- has a clause something like {their son wins the fight} as the direct object of helping.

As Daniel observes, help is unusual in that it doesn't require to, although it also doesn't forbid it. Both are grammatical, and there's no meaning difference; it's just stylistic -- that means some people will feel one is more formal than the other, but won't be able to agree on which one.

  • I helped him to win the prize.
  • I helped him win the prize.

Different people, in different contexts, may find some distinction here, but not much.

Some other verbs that can omit to before an infinitive object complement include

  • all modal auxiliaries (will, would, may, might, can, could, shall, should, must)
  • go and come in certain constructions (go be his assistant, come sit by me)
  • do in all its uses:
    • do-support do: Do you own that house?
    • emphatic do: He does own that!
    • active do: What he does is (to) read books.

Solution 5:

The plain form of the verb is preceded by the particle to in most instances where it follows another verb, so we would have to say, for example, Encouraging you to master Russian and not *Encouraging you master Russian. After the verb help, however, to is optional, and after some other verbs it is even disallowed. We cannot say *Making you to master Russian and we cannot say *Letting you to master Russian*.

The particle to is not really part of the verb at all. Not only is it not required after help, it is not allowed at all following modal verbs, or make, see, hear and let.