Jutro będziemy kibicować tej drużynie na stadionie.

Because kibicować is an imperfective verb. In Polish, the future of imperfective verbs is usually made with:

będę / będziesz / będzie / będziemy / będziecie / będą + infinitive

So:

będziemy kibicować = we will be cheering / we will cheer

This is a very common and natural pattern in Polish.

Polish often drops subject pronouns because the verb ending already shows who is doing the action.

Here, będziemy clearly means we will, so my is unnecessary.

You could say My jutro będziemy kibicować..., but that would usually add emphasis, as in we will cheer, not someone else.

Because the verb kibicować takes the dative case.

A useful pattern is:

kibicować komu? czemu?
= to cheer for / support + dative

The basic form is ta drużyna.
But in the dative singular it becomes:

• ta → tej
• drużyna → drużynie

So tej drużynie is the correct form after kibicować.

Tej is a form of the demonstrative ta.

In this sentence, it means something like this or that, depending on context. Polish does not always draw the same clear this/that distinction that English does.

So tej drużynie can correspond to:

• this team
• that team

If Polish really wants to stress that one over there, it can use tamtej drużynie.

Yes, that is grammatical.

Without tej, the phrase is less specific:

• drużynie = to a team / to the team, depending on context
• tej drużynie = to this/that team

So tej helps point to a particular team already known in the conversation.

Because na stadionie expresses location: being at the stadium.

After na, Polish uses:

• locative for location: na stadionie = at/in the stadium
• accusative for movement toward a place: na stadion = to the stadium

So:

• Jesteśmy na stadionie. = We are at the stadium.
• Jedziemy na stadion. = We are going to the stadium.

In your sentence, the cheering happens there, so location is needed: na stadionie.

Because Polish has no articles like a and the.

That means a noun like stadionie can mean:

• in a stadium
• in the stadium

The exact meaning comes from context.

Sometimes words like ten / ta / to help make something more specific, but they are not the same as English articles. In this sentence, tej adds specificity to drużynie, but stadionie has no article at all.

Putting jutro first is very natural because it sets the time frame right away.

It is similar to starting an English sentence with Tomorrow...

Polish often puts time expressions early in the sentence, especially in neutral statements:

• Jutro będziemy kibicować...

This sounds very normal and clear.

Yes. Polish word order is fairly flexible because endings show grammatical relationships.

All of these are possible, though they may sound slightly different in emphasis:

• Jutro będziemy kibicować tej drużynie na stadionie.
• Będziemy jutro kibicować tej drużynie na stadionie.
• Tej drużynie będziemy jutro kibicować na stadionie.
• Na stadionie będziemy jutro kibicować tej drużynie.

The original sentence is a very neutral, natural order: time → verb phrase → indirect object → place

No. With imperfective verbs, Polish can also form the future with będziemy + a past-like form:

• będziemy kibicować
• będziemy kibicowali
• będziemy kibicowały

The infinitive version będziemy kibicować is very common and neutral.

The other version shows gender/virility in the plural:

• kibicowali = a mixed group or a group of men
• kibicowały = a group of women or non-masculine plural

That is one reason learners often see and use the infinitive version first.

A rough pronunciation guide is:

Jutro – YOO-tro
będziemy – roughly ben-DZIE-my
kibicować – roughly kee-bee-CO-vatch
tej – like tay
drużynie – roughly droo-ZHI-nye
na stadionie – roughly na sta-DYO-nye

A useful general rule: Polish stress is usually on the next-to-last syllable.

So the main stresses are:

• JU-tro
• bę-DZIE-my
• ki-bi-CO-wać
• dru-ŻY-nie
• sta-DIO-nie

That rule works well for this whole sentence.