Haec epistula mihi omnium gratissima est.

Because haec has to agree with epistula.

Here epistula is feminine singular, so haec must be the feminine singular form of hic, haec, hoc, meaning this. In isolation, haec can be ambiguous, but the noun next to it tells you which form it is.

So haec epistula = this letter.

Epistula is nominative singular, and it is the subject of est.

The sentence is an X is Y type of sentence:

Haec epistula = the subject, this letter
gratissima = predicate adjective, describing the subject
est = is

If macrons were written, the ablative singular would be epistulā, not epistula. So in this sentence, the form is nominative singular.

Because Latin uses the dative with gratus.

Mihi is the dative singular of ego, and here it means to me. Latin says that something is pleasing/welcome to someone, not that it directly acts on them as a direct object.

So:

mihi gratissima est = is most pleasing to me / is dearest to me

An English speaker often expects something more like me, but Latin wants the dative here.

No, not in this sentence.

The adjective gratus can be tricky because it can be connected with the idea of gratitude, but when it describes a thing such as a letter, it means pleasing, welcome, dear, or acceptable.

A letter cannot really be grateful, so gratissima here means something like:

most pleasing
most welcome
dearest

So the sense is that this letter is especially welcome or dear to the speaker.

Because it agrees with epistula.

Latin adjectives must agree with the nouns they describe in gender, number, and case. Since epistula is feminine singular nominative, gratissima is also feminine singular nominative.

This is true even though mihi sits between them. The adjective still belongs with epistula.

Yes. Gratissima is the superlative form of gratus.

The forms are:

gratus = pleasing, dear
gratior = more pleasing, dearer
gratissimus / gratissima / gratissimum = most pleasing, dearest

So gratissima does not just mean pleasing. It means most pleasing or dearest. In English, a natural translation might even be my favorite in the right context.

Omnium is the genitive plural of omnis, meaning of all.

With a superlative, Latin often uses the genitive to show the group within which the comparison is being made. This is often called the genitive of the whole or partitive genitive.

So:

omnium gratissima = the most pleasing of all

English also uses of all, so this is a nice place where the Latin structure feels fairly familiar.

The noun is not stated, because it is understood from the context.

Latin often leaves out a noun when it is obvious. Depending on context, omnium could mean:

of all letters
of all things
of all the ones mentioned

Because the subject is this letter, many readers naturally understand something like of all letters, but the broader context would decide exactly what is meant.

Because Latin word order is flexible, and putting est at the end is very normal.

Latin relies much more on endings than on fixed word order, so the sentence can be arranged for emphasis or style. Here the order gives a smooth progression:

Haec epistula — the topic first
mihi — who it matters to
omnium gratissima — the important description
est — the final verb

English usually needs a more fixed order, but Latin does not. Ending with est is completely natural.