This law was just the last iteration in a long history of various legislations de repetundiis (concerning extortion/bribery). Cases were heard in a special "permanent" court, a quaestio perpetua, which had (probably) been established back in 149 BCE by the tribune L. Calpurnius Piso de pecuniis repetundiis (about provincial extortion). Cicero tells us as much in his Brutus 106. These quaestiones were fairly new in the mid-2nd century BCE and are interesting because they ushered in the era of professional prosecutors. Before them, major cases had been heard in the Assembly, where only a magistrate could prosecute.

We get the impression that this early quaestio had no teeth, and if it was designed to offer succor to wronged provincials, it wasn't very good at it. The jury was small and entirely drawn from the Senatorial order. There was apparently no penalty for conviction besides a requirement for restitution (a litis aestimatio, or assessment of damages). This changed with the next iteration, during the time of the Gracchi brothers (typically called the "lex Iunia"). Penalties became more severe and any provincial person was able to initiate the proceedings, not just Roman citizens via a patron. The jury was changed to be a panel of non-senators, a topic which is too big to broach here. Entire books have been written about the make-up of Roman juries in the Late Republic, and it was a frequent political hot potato. In addition to the quaestio about extortion, there were by the time of the Gracchi other quaestiones, like one for murder and one for corruption (ambitus).

There were further changes in a lex Servilia of about 104 BCE and then again by Sulla's reform in 81. Sulla returned the court's jury to the senators. Sulla is probably the one who brought the total number of permanent courts to seven, and stipulated that each have a presiding praetor. Since there weren't enough praetors for that, a iudex quaestionis stood in (probably an ex-praetor or other magistrate). Possibly at this time also was the division of the court into two sessions (actiones), which was apparently unique. In 70, L. Aurelius Cotta brought further refinement to the jury system with the introduction of "panels" (decuriones), one for senators, one for equestrians, and one for tribuni aerarii. The jurors were chosen by lot together from these three panels, 25 from each for the questio de repetundiis.

By the time we get to Caesar's sweeping law(s), the penalty de repetundiis in the quaestio had gained the option to become "capital." In Roman terms, this simply meant that a successful prosecution sent the defendant into exile. There was never a death penalty. By 59 also the court now apparently dealt with judicial corruption/bribery/extortion. This might at first seem odd, but recall that the court was designed to regulate provincial governors who had exploited their provincial subjects. It would follow that these defendants had gained a large sum of money from such activities and could therefore easily bribe the jury. And of course Caesar's law de repetundiis now greatly clarified or expanded what that term actually meant. It placed numerous prescriptions on Roman provincial magistrates which I won't enumerate here. The enforcement of these new parameters were still within the purview of the questio perpetua de repetundiis, however.

This is a brief rundown of the history of the court the various refinements. Its enforcement did not change after Caesar's departure, and in fact continued on into the early Principate with largely the same parameters. The quaestiones slowly became obsolete in the later Empire, but I do not myself know the details of that decline or what replaced them.

For much more details on various aspects of this, see Cloud's older but excellent overview of "The constitution and public criminal law" in CAH IX. In particular, he discusses the implications of the Gracchan era law's implications. The lemmata in the Oxford Classical Dictionary are really good for this topic, also, especially "QUAESTIONES* and REPETUNDAE.