Arrangören skickar in anmälningen för att deltagaren ska kunna delta.

Because skicka in is a particle verb meaning submit / send in.
On its own, skicka just means send.

In a main clause, Swedish puts the finite verb early in the sentence, so the particle often comes later:

• Arrangören skickar in anmälningen.

But in the infinitive, it stays together:

• att skicka in

Here, för att introduces a purpose clause. It means so that or in order for.

So:

• för att deltagaren ska kunna delta

means something like:

• so that the participant can / will be able to participate

It tells you why the organizer is submitting the registration.

Because the subject changes.

• Arrangören is the subject of skickar in
• deltagaren is the subject of ska kunna delta

If the same person did both actions, Swedish could leave out the second subject:

• Hon tränar för att kunna vinna.
  She trains to be able to win.

But here it is the organizer who submits something, and the participant who participates, so deltagaren must be stated.

Because this is about an intended result, not a simple present fact.

• kan delta = can participate right now / as a fact
• ska kunna delta = will be able to participate / is supposed to be able to participate

After för att, Swedish often uses ska to show what the action is meant to make possible.
So ska kunna delta fits the idea of purpose very well.

This part is a subordinate clause. In Swedish subordinate clauses, the normal order is:

subject + finite verb + infinitive(s)

So here:

• deltagaren = subject
• ska = finite verb
• kunna = infinitive
• delta = infinitive

That is why you get:

• för att deltagaren ska kunna delta

and not a main-clause-style word order.

The verb delta often appears with i when you say what someone is participating in:

• delta i tävlingen
• delta i kursen

But if the activity is already understood from context, Swedish can just use delta by itself.

So here, the idea is simply that the participant will be able to take part / participate, without naming the event again.

Both are definite singular forms, but the endings look different because the base nouns are different.

• arrangör → arrangören
• deltagare → deltagaren

Swedish usually adds definiteness as a suffix instead of using a separate word like English the.

So:

• arrangören = the organizer
• deltagaren = the participant

Yes.

• delta = to participate (verb)
• deltagare = participant (noun)

So deltagaren ska kunna delta may look repetitive to an English speaker, but it is perfectly normal Swedish:

• the participant should be able to participate

Yes, many learners notice that. In modern Swedish, anmälan is often the more common everyday word for registration / application / report depending on context.

Anmälningen comes from the related noun anmälning, which exists, but in many situations anmälan sounds more natural to many speakers.

So the sentence is understandable as written, but you will very often also see things like:

• skicka in anmälan
• skicka in sin anmälan