That's not what the constitution says though. "Government" ("Gobierno") is what an American would understand "executive branch" to be, I'm guessing this is why it's confusing. I tried to make it easier by adding the translations, but maybe that's just making it more confusing :)

I guess broadly in English you'd say the judges are part of the state, but they're not a part of the Spanish Government.

That's not how it works in Spain. In Spain, all members of the General Council of the Judiciary, which handles appointments to the Supreme Court, high courts, and other senior positions, are directly chosen by the Congress and the Senate after a reform by a socialist government in 1985.

This is against what the 1978 constitution says, but the Constitutional Court decided not to care.

You sum that to the president of the government bragging on live TV about how the current socialist government controls the prosecutors in Spain (https://www.youtube.com/watch?v=bbDsPfoE_a4) and you get a banana republic.