A few verbs use the preposition ” à “, a few others ” de ” and others nothing.
According to me, there aren’t specific rules, just various utilisations which some are idiomatic.
http://french.about.com/library/prepositions/bl_prep_verbs.htm
https://www.laits.utexas.edu/tex/gr/pre4.html
…

Here it means “to” (the “to” that forms part of an infinitive verb, such as to drive), and it functions as a way to link “essayer/try” to “vous montrer/show you”, and luckily this is how we do it in English also.

In general, “de” can mean a number of different things (https://fr.wiktionary.org/wiki/de), and much of it is grammatical and so makes sense only when considered with the words around it and its function in the sentence. But on its own, if often translates to “of” or “from”. “De” is quite versatile and as a result, unfortunately, confusing. Myself I’m learning French as well and still struggle with when/where to use it.

(As answered in comments by @Laure)
There is a difference between :

• try something / essayer quelque chose
  

  I tried this new restaurant. / J’ai essayé ce nouveau restaurant.

And

• try to do something / essayer de faire quelque chose
  

  I tried to not eat too much. / J’ai essayé de ne pas trop manger.

You added another question in comment between want/vouloir and try/essayer, the answer is different since it is more about “Why want to + [verb] = vouloir + [verbe]?”.

It is about being a modal verb : vouloir is, but not want.

Some other English verbs express modality although they are not modal verbs because they are not auxiliaries, including want[…] –Wikipedia