Tässä uudessa sanassa on kolme tavua.

Tässä is the inessive form of tämä, meaning this. In this sentence, it means in this.

So:

• tämä = this
• tässä = in this / here in this

In tässä uudessa sanassa, it helps form the meaning in this new word.

Because they all belong to the same phrase and are all in the inessive case, which usually means in.

Finnish shows this kind of agreement clearly:

• tämä → tässä = in this
• uusi → uudessa = in a new
• sana → sanassa = in a word

Together:

• tässä uudessa sanassa = in this new word

So unlike English, Finnish often marks the whole noun phrase with case endings, not just one word.

Sanassa is the inessive singular of sana.

• sana = word
• sanassa = in the word / in a word

The inessive ending is -ssa / -ssä, and it usually expresses being inside something.

Because uusi is one of those Finnish words whose stem changes in other forms.

Its basic pattern is:

• uusi = new
• uuden = of the new
• uudessa = in the new

So the stem changes from uusi- to uude- / uude-ssa, which gives uudessa.

This is a normal pattern for many -si words in Finnish. Another common example:

• vuosi = year
• vuoden = of the year
• vuodessa = in the year

This is a very common Finnish pattern in existential sentences.

The structure is:

• place/topic + on
  • thing that exists

So:

• Tässä uudessa sanassa on kolme tavua.
• literally: In this new word is three syllables.

Even though English uses there are, Finnish often uses singular on in this kind of sentence, especially when the thing after the verb is a quantity expression like kolme tavua.

Other examples:

• Pöydällä on kaksi kirjaa. = There are two books on the table.
• Huoneessa on yksi tuoli. = There is one chair in the room.

Because after numbers bigger than one, Finnish normally uses the noun in the partitive singular.

So:

• yksi tavu = one syllable
• kaksi tavua = two syllables
• kolme tavua = three syllables

This is one of the most important number rules in Finnish.

So kolme tavua is the correct form, not kolme tavut.

The dictionary form is tavu, which means syllable.

In this sentence:

• tavu = syllable
• tavua = partitive singular of tavu

It appears in the partitive because it follows the number kolme.

The sentence follows a very natural Finnish pattern:

• Tässä uudessa sanassa = location/topic
• on = is/are
• kolme tavua = what exists there

So the structure is roughly:

In this new word + there is/are + three syllables

A very literal translation would be:

• In this new word are three syllables.

But natural English usually says:

• This new word has three syllables.
• or There are three syllables in this new word.

It can mean either, depending on context.

On its own, tässä often means here:

• Olen tässä. = I am here.

But before a noun phrase, it can work like in this:

• tässä sanassa = in this word
• tässä kirjassa = in this book

So in Tässä uudessa sanassa, the best understanding is in this new word, not just here.

Because Finnish has no articles.

English uses a/an and the, but Finnish does not. That information is understood from context.

So:

• sana can mean a word or the word
• sanassa can mean in a word or in the word
• tässä sanassa specifically gives in this word

The demonstrative tässä makes the meaning definite enough, so no article is needed.

Yes. Finnish word order is flexible, though some versions sound more neutral than others.

Neutral:

• Tässä uudessa sanassa on kolme tavua.

Also possible:

• Uudessa sanassa on kolme tavua. = There are three syllables in the new word.
• Tässä sanassa on kolme tavua. = There are three syllables in this word.

A reordered version like:

• Kolme tavua on tässä uudessa sanassa

is possible, but it sounds more marked or emphatic. The original sentence is the most natural neutral form.