Arvum post villam est.

Because arvum is a neuter noun of the second declension. In Latin, many second-declension neuter nouns have -um in the nominative singular, which is the case normally used for the subject.

So even though an English speaker might not expect a subject to look like arvum, that form is completely normal for a neuter subject in Latin.

Because post takes the accusative case. The basic form of the noun is villa, but after post it must become villam.

So:

• villa = the dictionary form
• villam = accusative singular, used after post

It can mean either one, depending on context.

• In a time expression, post often means after
• In a place expression, post often means behind

Here, since the sentence is telling you where something is, post villam is understood spatially: behind the house/farmhouse

In Latin, case endings usually tell you what a word is doing in the sentence.

• arvum is nominative singular, which is the normal case for the subject
• villam is accusative singular, and here it is the object of the preposition post

So arvum is the thing being talked about, and post villam tells you its location.

Latin word order is much more flexible than English word order. Very often, especially in simple sentences, the verb comes at the end.

So Arvum post villam est is a normal Latin order.

Other orders are also possible, such as:

• Arvum est post villam
• Post villam arvum est

The meaning stays basically the same, though the emphasis can shift a little.

Latin does not have words that work exactly like English the and a/an.

So:

• arvum can mean the field or a field
• villam can mean the house/farmhouse or a house/farmhouse

You choose the best English article from the context.

Est is the third-person singular present form of esse, the verb to be.

So est means is.

That is why the sentence uses est rather than a plural form like sunt.

Yes, when post is used as a preposition, it normally takes the accusative case.

That is why you get post villam, not post villa.

This is something worth memorizing: some Latin prepositions regularly govern a particular case, and post is one of the common ones that takes the accusative.