Learn Hardcore Swedish: Jag säger att jag inte har bråttom, eftersom jag inte vill bli stressad. - I say that I’m not in a hurry, because I don’t want to become stressed.

Att introduces a subordinate clause that functions as the object (the “content”) of säger.
So Jag säger att … = “I say (that) …”. In Swedish, you very often keep att in writing and in careful speech, more often than English keeps “that”.

Why is jag repeated: Jag säger att jag …? Can it be omitted?

It’s repeated because the clause after att is its own clause with its own subject. Swedish generally requires an explicit subject in finite clauses.
You normally can’t drop the second jag the way some languages do.

Why is the word order jag inte har (not inte jag har)?

In a main clause, Swedish has verb-second (V2) word order, and inte usually comes after the finite verb: Jag har inte ….
But in a subordinate clause (like after att), the negation typically comes before the finite verb: … att jag inte har ….
So this sentence shows the common pattern: subordinate clause = subject + inte + verb.

What does ha bråttom literally mean, and why is it phrased with har?