Under what authority did they seemingly take control of the eastern half of the Republic and gather these thousands of soldiers? When I read about it, it seems so sudden given that it all happened within two years of Caesar's death.
I apologize if this is a short answer, but it's not a complex question. Brutus and Cassius were both praetors in 44, and in 43 took provinces as proconsuls. Cassius was proconsul in Syria, and Brutus apparently was proconsul in several provinces - Illyricum, Macadonia, and Achaea, and maybe later acquired Asia as well. This was agreed on by Antony at Cicero's proposal for a settlement after Caesar's assassination: Caesar's will was ratified as well as his disposition of the government, and an amnesty was granted for the assassins. These promagisteries were extended for 42.
So that was their authority: they were proconsuls of Rome.
That settlement dissolved after Cicero turned on Antony and used Octavian to try to prevent him from taking his province in Gaul. Instead, Octavian and Antony and Lepidus entered into an agreement for taking control of the state, the Second Triumvirate, and this deal included tracking down and killing the assassins.
It kinda illustrates how civil this civil war was. Octavian was suffect-consul. Antony was a proconsul, as were Brutus and Cassius. These were all legitimate offices in Roman government. It was really the state fighting itself - there are no rebels here.