Na deser kroję arbuza i ananasa na małe kawałki.

Because Polish often drops subject pronouns when they are already clear from the verb form. The ending -ę in kroję tells you it is 1st person singular, so I cut / I am cutting.

You could say Ja kroję..., but that would usually add emphasis, as in I’m the one cutting it.

This is an aspect question.

• kroić = imperfective
• pokroić = perfective

Kroję describes an action as ongoing, repeated, or simply viewed as a process:
I am cutting / I cut

Pokroję is perfective, so it usually refers to a completed action in the future:
I will cut up

So in this sentence, kroję is natural if the speaker is describing what they are doing now, or giving a recipe-like description.

They are the direct objects of the verb kroję, so they must be in the accusative case.

For many masculine nouns referring to food items, the accusative singular takes the ending -a:

• arbuz → arbuza
• ananas → ananasa

This may seem surprising because these things are not alive, but Polish treats many masculine food words with this animate-like accusative pattern.

Here they are accusative.

That is a very common learner question, because for these nouns the accusative singular looks exactly like the genitive singular:

• nominative: arbuz
• genitive: arbuza
• accusative: arbuza

You know it is accusative here because the words are functioning as the direct objects of kroję.

Na deser is a very common expression meaning for dessert or as dessert.

Here, na does not mean a literal on. It introduces the role or purpose of something:

• na obiad = for lunch
• na kolację = for dinner
• na deser = for dessert

So the phrase tells you when or in what role the fruit is being served.

It actually is in the accusative, but you do not see a different form.

The noun deser is masculine singular, and for this noun:

• nominative: deser
• accusative: deser

So the case changes grammatically, but the form stays the same.

This second na has a different job. It means something like into in English when you divide or transform something.

So:

• kroić coś na kawałki = to cut something into pieces
• pokroić coś na plasterki = to slice something into slices
• podzielić coś na pół = to divide something in half

So na małe kawałki means into small pieces.

Because the phrase na małe kawałki uses the accusative plural.

• singular: mały kawałek
• plural: małe kawałki

For masculine inanimate nouns like kawałek, the accusative plural has the same form as the nominative plural. That is why you get:

• kawałki
• małe kawałki

Not małych kawałków, because that would be a different case.

No, Polish word order is fairly flexible. This version is natural because Na deser is placed first to set the context.

The sentence could be rearranged, for example:

• Kroję arbuza i ananasa na małe kawałki na deser.

That still works, but the original sounds smoother and more natural in many contexts. In Polish, word order often changes emphasis rather than basic meaning.

Polish has no articles, so there is no direct equivalent of a or the.

That means arbuza can mean:

• a watermelon
• the watermelon

and ananasa can mean:

• a pineapple
• the pineapple

Context tells you which is meant. In a real situation, listeners usually understand this easily from what has already been mentioned or what is physically present.

It can mean either, depending on context.

Polish present tense does not make the same sharp distinction that English does between:

• I cut
• I am cutting

So kroję can describe:

• an action happening right now
• a habitual action
• a recipe or narration style

In this sentence, the most natural reading is often I’m cutting watermelon and pineapple into small pieces for dessert, but I cut watermelon and pineapple into small pieces for dessert is also possible in the right context.