Otium sine libro mihi dulce non est, sed avia sub arbore quiete sedere amat.

Because sine always takes the ablative in Latin.

So:

• sine libro = without a book
• libro is ablative singular of liber

This is something you simply learn as part of the preposition:

• sine + ablative = without
• compare cum + ablative = with

Mihi is the dative singular of ego, meaning to me or for me.

In this sentence, it shows the person for whom something is pleasant:

• mihi dulce non est = is not sweet/pleasant to me

This is very natural in Latin. Instead of saying something like I do not find leisure sweet, Latin often says:

• otium mihi dulce est = leisure is pleasant to me

This is often called a dative of reference or dative with adjectives like dulcis.

Because dulce agrees with otium, and otium is neuter singular.

The adjective dulcis, dulce has:

• masculine/feminine nominative singular: dulcis
• neuter nominative singular: dulce

Since otium is a neuter noun, the predicate adjective must also be neuter:

• otium ... dulce est = leisure is pleasant

Otium is more than just free time in a modern casual sense. It often means:

• leisure
• rest
• peaceful time away from work
• sometimes even cultivated relaxation

So otium sine libro suggests something like leisure without a book, not just spare time in a vague sense.

It is a very Roman word, often contrasted with negotium (business, work, public activity).

In Latin, non usually negates the verb or the whole statement, and its position is more flexible than in English.

So:

• mihi dulce non est
• literally: to me sweet not is
• naturally: is not pleasant to me

Latin word order is often arranged for emphasis or style rather than fixed grammatical necessity. Here the sentence is perfectly normal.

If you wrote mihi non dulce est, that could also work, but the given order is very idiomatic.

Yes, avia normally means grandmother.

It is the feminine noun corresponding to avus (grandfather).

A beginner might confuse it with similar-looking words, but here:

• avia = grandmother

It is nominative singular in this sentence and is the subject of amat.

Because sub can take either the accusative or the ablative, depending on the meaning.

• sub + ablative = under, in a stationary location
• sub + accusative = to under, movement toward a position under something

Here the grandmother is already sitting in that location, not moving there, so Latin uses the ablative:

• sub arbore = under a tree

Quiete is being used adverbially and means quietly, calmly, or peacefully.

So:

• quiete sedere = to sit quietly

This is a good example of Latin using a form that looks like an ablative to express manner. For a learner, the easiest way to understand it here is simply as an adverbial word modifying sedere.

Because Latin often uses an infinitive after verbs like amo to express liking/loving to do something.

So:

• sedere amat = she loves to sit

This is very similar to English:

• She loves to sit under a tree
• Latin: sub arbore ... sedere amat

The infinitive sedere means to sit.

The second half is:

• sed = but
• avia = subject, grandmother
• sub arbore = prepositional phrase, under a tree
• quiete = adverbial word, quietly
• sedere = infinitive, to sit
• amat = main verb, loves

So the structure is essentially:

• But grandmother loves to sit quietly under a tree

Latin allows these elements to appear in an order that is more flexible than English.

Because Latin has no articles.

So Latin does not have separate words for:

• a / an
• the

Whether you translate libro as a book or the book, and arbore as a tree or the tree, depends on context.

In this sentence:

• sine libro = without a book
• sub arbore = under a tree

Those are the most natural English translations, even though Latin does not explicitly mark the difference.

You know from the forms and the clause structure.

In the first clause:

• otium is nominative singular, so it is the subject
• est is singular, matching it
• dulce agrees with otium

In the second clause:

• avia is nominative singular, so it is the subject
• amat is singular, matching it
• sedere is an infinitive depending on amat

Latin relies much more on case endings and verb forms than on strict word order. That is why you can still identify the grammar even when the words are not arranged like English.