Questions & Answers about Я написал вам пять писем.
Why does the word for "letters" (писем) look so different from its dictionary form (письмо)?
Because it is in the genitive plural! Neuter nouns ending in -о (like письмо) form the genitive plural by dropping the -о entirely. This is called a zero ending.
If we just drop the -о from письмо, shouldn't it be письм? Where did the е in писем come from?
Russian avoids ending words with awkward clusters of consonants. When dropping the -о leaves a hard-to-pronounce clump at the end, a fleeting vowel (usually о or е) is inserted to make it easier to say. Here, an е is added, and the soft sign (ь) is dropped because the е already does the job of making the previous consonant soft.
Why do we need the genitive plural for "letters" here anyway?
It is because of the number пять (five). In Russian, exact quantities of 5 and above require the noun that follows them to be in the genitive plural.
Why is "you" translated as вам instead of вы or вас?
вам is the dative case of вы. The dative case is used for the indirect object, which is the person receiving the action. Since you are writing to them, they are the recipient of the letters, so we use the dative case.