Этот штат большой.

Because штат is a masculine singular noun.

In Russian, the demonstrative this changes to match the noun’s gender, number, and case:

• этот = masculine singular
• эта = feminine singular
• это = neuter singular
• эти = plural

Since штат is masculine, you say этот штат.

It is in the nominative case.

In Этот штат большой, штат is the subject of the sentence, so nominative is the normal case to use.
The adjective этот also appears in the nominative masculine singular to match it.

Because adjectives in Russian must agree with the noun they describe.

Here, большой agrees with штат, which is:

• masculine
• singular
• nominative

So the adjective takes the masculine singular nominative form большой.

Compare:

• большой штат = a big state
• большая страна = a big country
• большое озеро = a big lake
• большие города = big cities

In Russian, the verb to be is usually omitted in the present tense.

So:

• Этот штат большой. = This state is big.

Russian does not normally say a separate word for is in sentences like this.

Compare:

• Он врач. = He is a doctor.
• Дом большой. = The house is big.

In past and future tense, forms of to be do appear:

• Этот штат был большим. = This state was big.
• Этот штат будет большим. = This state will be big.

It is an adjective, used as a predicate adjective.

In English, this works like big in The state is big.
It describes the subject штат, but it comes after the noun because the sentence structure is basically:

• This state — big.

That is a very normal Russian pattern.

Yes. Russian word order is more flexible than English word order.

The neutral, standard order is:

• Этот штат большой.

But you may also see:

• Штат этот большой.
• Большой этот штат.

These alternatives can change the emphasis or sound more marked. For a learner, Этот штат большой is the safest and most natural basic version.

A simple pronunciation guide is:

• этот ≈ EH-tuht
• штат ≈ shtaht
• большой ≈ bal-SHOY

The stress is on:

• э́тот
• штат (one syllable)
• большо́й

A rough full pronunciation:

• EH-tuht shtaht bal-SHOY

Note that unstressed vowels in Russian are often reduced, so о in большой does not sound like a clear English o.

Not always.

Штат can mean:

1. state, especially a state in a country such as the US
2. staff/personnel, in some contexts

In this sentence, because of the meaning you were given, it clearly means state.

So here:

• Этот штат большой. = This state is big.

But in another context, штат could refer to employees or staffing.

Both can sometimes be translated as big or large, but they are not always interchangeable.

• большой is the most general and common word for big/large
• крупный often means large-scale, major, or significant

For physical size or general size, большой is the normal beginner word.
So Этот штат большой is the straightforward, natural way to say This state is big.

Not if you mean This state is big.

Это by itself usually means this is / it is / this as a neuter demonstrative, and it does not work here as the adjective modifying штат.

You need этот because it directly modifies the masculine noun штат:

• Этот штат большой. = This state is big.

By contrast:

• Это большой штат. = This is a big state.

That is a different sentence structure.

They are both natural, but they mean slightly different things structurally.

• Этот штат большой. = This state is big.
• Это большой штат. = This is a big state.

In the first sentence:

• этот штат is the subject
• большой is what is being said about it

In the second sentence:

• это means this is
• большой штат is the thing being identified

English often treats these as very similar, but in Russian they are built differently.