Roman governors were promagistrates, particularly proconsuls, and as such they retained the imperium of their magistracy in the provinces. Actually, the power of a proconsular governor was effectively greater than that of many consuls, as it usually lasted five years instead of one and the proconsul was the sole magisterial authority of the province. Add in the fact that proconsuls could not be prosecuted during their term of office (though they could be recalled by order of the senate, which legally the governor did not have to listen to at all) and the fact that they were provided with an unlimited number of legions for a full five years (consular armies were only four legions, and even in the unlikely event that a consul was at-arms his whole term, that meant only a year) and proconsular governors became pretty powerful. They had the right to set up judicial proceedings, order special taxes, levy auxiliaries, etc. A governor's province was also expected to shoulder most of the cost of his army and administration through taxes. In theory a governor was supposed to keep records of his expenses and everything, but there were easy ways around that. Cicero's prosecution of Piso gives us good insight as to the most common means of extortion and laundering. In provinces like Piso's Macedonia that had large armies the cost of the soldier's equipment was paid for partially by the state and also by the province--Cicero accuses Piso of writing troops that didn't exist into his records so that he could pocket the money supplied to him for equipment. Additionally, the food and upkeep of the troops was the duty of the province--governors levied a special tax on foodstuffs and in coin, and the tax was often inflated. Various taxes and payments were all relatively legal ways to do it, but governors also had a number of less kind methods if they really wanted to strike it rich. In provinces like Syria, Macedonia, or Gaul that had large armies and were under constant threat of invasion the reduction of enemy cities was an easy way to make money--in addition to the spoils from the city itself there were slaves and goods that could be seized, as well as trade lanes. Caesar's unimaginable wealth from Gaul appears mainly to have been in slaves and physical spoil, not in the financial tricks which many governors played. Caesar, because of his incredible success and more or less limitless power during his proconsulship, is something of an exception, but we see things like this all the time. Piso was accused of sacking his own cities in order to take their stuff and sell them as slaves (an absurd accusation--Piso's proconsulship actually appears to have been exemplary, and he even disbanded his army according to law when his campaigns were over, unlike many governors). Piso was also accused of overrunning Roman allies and friends to sack their cities and put the money in his pocket (also absurd--the various people Cicero calls "friends of Rome" had repeatedly attacked the Romans several times in the last century and appear to have attacked Piso first, not the other way around). Not to mention all the tribute that could be exacted from subject peoples, which was levied at the whim of the governor and often collected by military forces

Governors also, as sole magisterial authority of the province, exercised enormous legal power, in addition to their military and administrative powers. Even a lowly propraetorian governor like Verres had the power to confiscate property and bring people to trial before him, trials which were often very unfair for obvious reasons. Verres levied multiple new grain taxes, confiscated great works of Greek art from private individuals (some of them citizens, or from temples), and brought a number of Roman citizens to trial or had them outright executed (this last illegally). Verres also had a trick where he would confiscate enormous landholdings by accusing the slaves working estates of being partisans of Spartacus, and would then throw the owner into prison if he didn't provide him with a sizeable bribe, confiscating his land in the process. Of course, Verres was prosecuted when he returned, but not directly for anything that he did to the provincials. That made everything worse, sure, and Cicero used it to prove that Verres had totally mismanaged his province, but the only real law he broke was his execution of Roman citizens without trial and his general disregard for the rights of Roman citizens. The injustice done to the Sicilian provincials was bad, and evidence both that Verres was an ass and that he was a lousy fucking governor, but it wasn't actually illegal. Piso was accused of a similar thing, confiscating the property of a local Macedonian and having him executed when he resisted.

Governors also had some less nasty ways of acquiring great wealth. In provinces like Macedonia, where there was a large number of free cities, gifts from the inhabitants of free cities were not only possible, but expected. Piso received all kinds of honors from the free Greek cities, including vast sums of money or enormous quantities of grain

It's important to note, however, that not all Roman governors did this stuff--in fact, most appear to have been perfectly normal. Catullus writes a poem addressed to some of his friends about how the governor of Bithynia he's serving under is a cheapskate and doesn't provide him with any opportunities to get rich, for which he gets laughed at in another poem, and implies that his friends (serving under Piso in Macedonia!) are in a similar boat, under a responsible and upright governor. Even so, a proper governor could still make huge amounts of money, more or less by accident. The funds provided to a governor for various expenses were regularly way more than was needed, and tribute and gifts easily padded out a governor's pocketbook even if he didn't set out to wring out more money by force.