Kajakarz wywraca się w bystrzu.

Here się is part of the verb expression wywracać się.

It does not literally mean that the kayaker is doing something to himself. Instead, it helps make the verb mean to overturn / to capsize / to tip over.

• wywracać = to turn something over, overturn something
• wywracać się = to overturn, tip over, capsize

So in this sentence, się is essential. Without it, the meaning would change.

Because the intended meaning is the kayaker capsizes / overturns, not the kayaker overturns something.

Compare:

• Kajakarz wywraca kajak. = The kayaker overturns the kayak.
• Kajakarz wywraca się. = The kayaker overturns/capsizes.

So się turns it into the intransitive meaning.

Wywraca się is present tense of the imperfective verb wywracać się.

That means it can express things like:

• something happening right now
• something that happens repeatedly
• a vivid present used in storytelling

Depending on context, it could correspond to English:

• is capsizing
• capsizes
• keeps capsizing
  though English usually chooses one of these based on context

The perfective partner is wywrócić się. For example:

• Kajakarz wywróci się. = The kayaker will capsize.
• Kajakarz wywrócił się. = The kayaker capsized.

Kajakarz is in the nominative singular, because it is the subject of the sentence.

It is the person doing or undergoing the action:

• kajakarz = kayaker
• nominative singular masculine animate

This is also the dictionary form of the noun.

Because w means in here, and when w expresses location, it normally takes the locative case.

So:

• dictionary form: bystrze
• locative singular: bystrzu

That is why we get:

• w bystrzu = in the rapid / in the rapids / in whitewater

A useful comparison is:

• morze → w morzu
• bystrze → w bystrzu

So this -u ending is not random.

Because w bystrze would not be the normal form for location.

In this sentence, the phrase tells you where the capsizing happens, so Polish uses the locative:

• w bystrzu

If a preposition like w expresses movement into something, it can take the accusative, but that is not what is happening here. This sentence is about the place of the event, not motion into the rapid.

It is singular locative.

Even though English often says in the rapids, Polish can use a singular noun here. That is fairly normal.

If you wanted an explicitly plural form, you would use a different plural case form, for example w bystrzach. But w bystrzu is perfectly natural for this sentence.

Yes, sometimes się can move, because Polish word order is more flexible than English.

For example, you may hear:

• Kajakarz wywraca się w bystrzu.
• Kajakarz się wywraca w bystrzu.

Both are possible, though the first one sounds very natural and keeps the verb phrase together.

One important rule: się usually does not stand at the very beginning of a sentence.

No, Polish word order is fairly flexible.

The sentence as given is a neutral, natural order:

• Kajakarz wywraca się w bystrzu.

But you could rearrange it for emphasis, for example:

• W bystrzu kajakarz wywraca się.

That puts more attention on where it happens.

So the original order is not the only possible one, but it is a very normal default choice.

Because Polish does not have articles like English a and the.

So kajakarz can mean:

• a kayaker
• the kayaker

And w bystrzu can mean something like:

• in a rapid
• in the rapid
• in the rapids

Context tells you which interpretation fits best.

Sometimes, but wywraca się is especially natural for something overturning completely, such as a kayak, boat, or vehicle.

Very roughly:

• przewracać się = to fall over / tip over
• wywracać się = to overturn / flip over / capsize

In a kayaking or whitewater context, wywraca się is a very good choice.

A rough English-friendly guide:

• kajakarz ≈ ka-ya-kash
• wywraca ≈ vuh-vra-tsa
• się ≈ shyeng or shyeh depending on how exact you want to be
• bystrzu ≈ BIST-shoo

A few helpful points:

• rz at the end of kajakarz is devoiced, so it sounds like sz
• cz sounds like English ch, but harder
• w in Polish sounds like English v
• y is not the same as English ee

So the whole sentence is roughly:

ka-ya-kash vuh-vra-tsa shyeh f BIST-shoo

The w before bystrzu is pronounced like f here because of the following voiceless consonant sound.