Haec epistula mihi gratior est quam illud munus.

Because haec has to agree with epistula.

Epistula is:

• feminine
• singular
• nominative

So the demonstrative hic, haec, hoc must also be feminine singular nominative here, which is haec.

So:

• hic = masculine nominative singular
• haec = feminine nominative singular
• hoc = neuter nominative singular

Since epistula is feminine, Latin uses haec epistula = this letter.

For the same reason: agreement.

Munus is a neuter noun, so the demonstrative must be neuter too. The nominative singular neuter form of ille, illa, illud is illud.

So:

• ille = masculine nominative singular
• illa = feminine nominative singular
• illud = neuter nominative singular

Therefore:

• illud munus = that gift

A learner may expect a noun ending in -us to be masculine, but munus is different: it is a third-declension neuter noun.

Mihi is the dative singular of ego.

It is used because gratus and its comparative gratior commonly take the dative of the person to whom something is pleasing, welcome, or dear.

So the structure is not exactly I like this letter more in Latin grammar. It is more like:

This letter is more pleasing/welcome/dear to me than that gift.

That is why Latin uses:

• mihi = to me

rather than a nominative ego or an accusative me.

Gratior is the comparative form of gratus, grata, gratum.

The basic adjective gratus can mean things like:

• pleasing
• welcome
• dear

Its comparative gratior means:

• more pleasing
• more welcome
• dearer

So mihi gratior est means is more pleasing to me or is dearer to me.

This is a very common Latin comparative pattern:

• positive: gratus
• comparative: gratior
• superlative: gratissimus

Because the subject is haec epistula, which is singular.

Latin finite verbs agree with their subject in person and number. Since epistula is one letter, the verb is singular:

• epistula ... est = the letter is

Even though illud munus also appears in the sentence, it is part of the comparison, not a second subject requiring a plural verb.

Quam introduces the second thing in a comparison after a comparative adjective such as gratior.

So:

• haec epistula = the first thing being compared
• illud munus = the second thing being compared
• gratior ... quam = more pleasing than

A very literal analysis is:

This letter is more pleasing to me than that gift.

With quam, the second item is usually put in the same case as the first item being compared. Here the first item, haec epistula, is nominative, so illud munus is nominative too.

Not here. After a comparative with quam, Latin normally puts the second item in the same case as the first item compared.

Here the comparison is between:

• haec epistula — nominative
• illud munus — also nominative

So the structure is:

haec epistula ... gratior est quam illud munus

Literally, you can think of it as an abbreviated version of:

This letter is more pleasing to me than that gift is pleasing to me.

The second is pleasing to me is left understood.

Yes. Latin often has two ways to make this kind of comparison:

• quam
  • same case
• ablative of comparison without quam

So this sentence could also be written as:

Haec epistula mihi gratior est illo munere.

That means the same thing: This letter is more pleasing to me than that gift.

In your sentence, the author chose quam illud munus. That is often especially clear for learners, because the comparison is very explicit.

Yes, you can understand an omitted idea after it.

Latin often leaves out words that are easy to supply from the context. Here, after quam illud munus, you can mentally supply something like:

• est
• or even mihi gratum est

So the fuller sense is:

This letter is more pleasing to me than that gift is.

Latin does not need to repeat the whole predicate when the meaning is obvious.

Because Latin word order is much freer than English word order.

English depends heavily on position:

• This letter is dearer to me than that gift

Latin depends much more on endings:

• haec shows feminine nominative singular
• epistula is nominative singular
• mihi is dative
• gratior is comparative
• illud munus is the second item of comparison

So Latin can arrange the words for emphasis or style. The given order is perfectly natural. A very literal word-by-word order would be:

This letter to me dearer is than that gift.

That sounds odd in English, but it is normal in Latin.

Either can work, depending on context.

The Latin adjective gratus covers a range that English often splits into several words. So gratior might be translated as:

• more pleasing
• more welcome
• dearer

In this sentence, all of those are possible depending on the tone you want. If the context is emotional, dearer may sound right. If the context is more neutral, more pleasing or more welcome may fit better.

So the Latin grammar stays the same; only the English wording shifts slightly with context.