I’ll give you an example, in French, you’d say:

Il est allé marcher (he went for a walk)

and not

Il a allé marcher

Because ‘Aller’ belongs to the ‘Vandertrampp list’. In other words, you’d use the ‘être’ auxiliary verb, not the ‘avoir’.

To be honest I had never heard about that list, and I wouldn’t apply it to the letter.

For instance, the first M is supposed to stand for ‘Monter’. Although you can use ‘être’ in some cases, ‘Il est monté à Paris’ for instance, ‘être monté’ most of the time means… something totally different. In my opinion it’s dangerous to include verbs used with both auxiliary verbs in this list.

In passé composé, you need to choose an auxiliary verb, either aller or être.

French teachers usually simplify the rule and say: If a verb is on the DMV list, use être.

J’ai parlé → Not DMV. Use avoir.

Je suis sorti → DMV. Use être.

The full rule throws a small wrench in the gears. A DMV verb only uses être if it has no direct object. (Also, any verb can use être if it’s reflexive. Here’s a diagram.) Hence:

J’ai mangé une pomme → Not DMV, has an object. Use avoir.

J’ai parlé → Not DMV, has no object. Use avoir.

J’ai sorti mon livre → DMV, has an object. Use avoir.

Je suis sorti ce soir → DMV, has no object. Use être.

This is what vc74 means about « Il a monté les escaliers » (les escaliers is the object of monté).

The other part of your question is what makes these verbs special. This is a long-running question.

The closest way to generalize them is that it’s loosely related to state changes: birth, death, change in location… however, for every DMV verb about a change of state, there’s an alternative with a similar meaning that isn’t DMV.

At the end of the day, it’s a historical quirk — the semantic landscape it’s based on is much older than our current vocabulary, so it won’t fit perfectly, if it ever did.

In the following reference, "Fun With Grammar", you will the find essential information about this mnemonic device.

• It is meant to provide an easy way to remember which verbs of the intransitive sort are conjugated with "être": they are a few exceptions, as almost all such verbs are conjugated with the auxiliary verb "avoir". (reflexive verbs are all conjugated with "être").

• It is not an ideal list: there are more verbs than the list really mentions, and so you have to learn additional memorizing techniques. This makes this device more a nuisance than a useful means to learn.

• Il is better to learn through the practice of reading: the result is much better.

• (not mentioned in the reference, user LPH’s suggestion) In any case, the list and the explanations that go along with it, as in the particular reference used in the present answer, if kept in the proper place in your files, will provide a means of quick reference when you are writing (or reading).

There are more details and the reference should consulted for those.