Ŝi asertas, ke ŝi venkis.

Ke introduces a subordinate clause and means that.

So the structure is:

• Ŝi asertas = she claims / she asserts
• ke ŝi venkis = that she won

In Esperanto, ke is very commonly used where English may use that, even in places where English often leaves that out.

Because the two verbs describe two different times:

• asertas = she is claiming it now
• venkis = the winning happened earlier

Esperanto tense endings are very straightforward:

• -as = present
• -is = past
• -os = future

So Ŝi asertas, ke ŝi venkis means she is making the claim now, and the victory happened in the past.

Also, Esperanto usually does not do the kind of tense shifting English sometimes does. It simply states the time of each action directly.

Because Esperanto normally needs an explicit subject in each clause.

So:

• Ŝi asertas = she claims
• ŝi venkis = she won

In English, we sometimes leave things out more easily in certain structures, but in Esperanto the subject is normally stated clearly. Without the second ŝi, the sentence would sound incomplete.

Because sin is a reflexive pronoun used for an object, not for a subject.

Here, in the clause ke ŝi venkis, ŝi is the subject of venkis, so it must be ŝi.

Compare:

• Ŝi vidas ŝin. = She sees her.
• Ŝi vidas sin. = She sees herself.

But in your sentence, there is no reflexive object. The second ŝi is simply the subject of the subordinate clause.

In normal Esperanto punctuation, yes, a comma is usually placed before a subordinate clause introduced by words like ke, ĉar, se, and so on.

So:

• Ŝi asertas, ke ŝi venkis.

This is the standard way to write it.

A simple pronunciation guide:

• Ŝi = shee
• asertas = ah-SER-tahs
• ke = keh
• venkis = VEN-kees

A few helpful points:

• ŝ sounds like English sh
• stress in Esperanto is almost always on the second-to-last syllable
  • aSERtas
  • VENkis

So the full sentence sounds roughly like:

shee ah-SER-tahs, keh shee VEN-kees

Yes, but the given order is the most neutral and natural.

The basic structure is:

• main clause: Ŝi asertas
• subordinate clause: ke ŝi venkis

You could rearrange things in special contexts for emphasis, but Ŝi asertas, ke ŝi venkis is the standard way to say it.

Not exactly.

Aserti means assert, state firmly, or claim. It sounds stronger and a bit more deliberate than just say.

So:

• diri = to say
• aserti = to assert / claim

In many contexts, aserti suggests that the speaker is presenting something as true, perhaps confidently or insistently.

This is a very useful distinction.

• venki = to defeat, to overcome, to be victorious
• gajni = to win, gain, earn

Examples:

• Li venkis sian kontraŭulon. = He defeated his opponent.
• Li gajnis premion. = He won a prize.

But Esperanto also allows venki without a direct object, meaning to win / be victorious, so ŝi venkis is perfectly normal.

Very roughly:

• venki focuses on being victorious
• gajni often focuses on obtaining something

Because here venki is being used intransitively, meaning to win or to be victorious.

So:

• Ŝi venkis. = She won.
• Ŝi venkis la kontraŭulon. = She defeated the opponent.

Both are possible. In your sentence, the object is simply not stated.

Yes, but the ke version is simpler and more common for learners.

A possible alternative is:

• Ŝi asertas esti venkinta.

That literally means something like She claims to be having won, which corresponds to English She claims to have won.

However, the version with ke is usually more straightforward:

• Ŝi asertas, ke ŝi venkis.

That is the clearest and most natural basic pattern to learn first.