In campo alta arbor stat, et pueri sub umbra eius legunt.

Because in can take different cases depending on the meaning. Here it means in or on in the sense of location, so it takes the ablative case.
Campus means field, and its ablative singular form is campo. So in campo means in the field.

Because adjectives must agree with the nouns they describe in gender, number, and case.
Arbor is a feminine noun, singular, and here it is nominative because it is the subject. So the adjective must also be feminine nominative singular: alta.
Even though arbor ends in -or, it is still feminine.

Arbor is nominative singular. We know this because it is the subject of stat: the tree is the thing that is standing.
In Latin, the subject is usually in the nominative case. So alta arbor stat means the tall tree stands.

Stat specifically means stands or is standing, while est just means is.
So stat gives a more physical, visual idea: the tree is upright there in the field. Latin often chooses a more vivid verb where English might simply say is.
So alta arbor stat is more descriptive than alta arbor est would be.

Like in, the preposition sub can take more than one case. When it means under in a static sense, it usually takes the ablative. When it implies movement toward a position under something, it can take the accusative.
Here the boys are reading under the shade, not moving under it, so Latin uses the ablative: sub umbra.

Umbra is the ablative singular of umbra, meaning shade or shadow.
It is ablative because it follows sub in a location sense. Since umbra is a first-declension feminine noun, its ablative singular form is umbra.

Eius means his, her, its, or their, depending on the context. Here it means its, referring back to arbor, so sub umbra eius means under its shade.
Latin uses suus, sua, suum mainly when the possessor is the subject of the same clause. Here the shade belongs to the tree, but the subject of the second clause is pueri, not the tree, so eius is the natural choice.

Pueri is nominative plural, so it means the boys as the subject.
The verb legunt is also plural, so the forms match: pueri legunt = the boys read / are reading. If Latin wanted boys in another role, such as an object, it would use a different case form.

Latin, like English, does not always need to state the object if it is understood or unimportant.
So legunt can simply mean they are reading or they read without saying exactly what they are reading. The sentence focuses on the scene, not on the specific text.

It can mean either. The Latin present tense often covers both the English simple present and present progressive.
So pueri legunt can mean the boys read or the boys are reading. In this kind of descriptive sentence, are reading is often the most natural English translation.

Classical Latin does not have articles like English the and a/an.
So a noun such as arbor can mean a tree or the tree, depending on context. The same is true for pueri: it can mean boys or the boys.
English has to choose an article, but Latin usually leaves that to context.

Latin word order is much more flexible than English word order because the endings show each word’s role.
So In campo alta arbor stat could be rearranged in several ways without changing the basic meaning. However, the chosen order sounds natural and helps the sentence flow: place first, then the tree, then the verb; then the second clause with et.
Word order in Latin often adds emphasis rather than determining the basic grammar.

That is a common and natural position for an adjective, especially in a simple descriptive sentence.
Latin can put adjectives before or after nouns, but the choice may affect style or emphasis. Here alta arbor neatly presents the tree as tall right away.
So the position is not required in the same way as in English, but it is perfectly normal Latin.

Not really. Latin punctuation in modern editions is an editorial aid, not an original part of the ancient language in the same way as case endings and verb forms.
The comma here simply helps the reader see the two linked clauses: alta arbor stat and pueri sub umbra eius legunt. The important grammatical link is et, meaning and.