How do tax farmers work? I understand they are assigned a region and promise the king they will produce a certain amount of revenue with everything past that being theirs to keep. However I don’t understand the actual process. Do they bring troops with them to help shake down the people? Were the taxes just arbitrary? Was there a tax season? How did the smallfolk feel about this? Merchants? Nobles? 

You’ve kind of missed a step in the process that goes a length to explaining why the system was so messed up. The promise in question was usually an up-front lump-sum payment given in exchange for the right to the ongoing revenue-stream in that province. Yes, sometimes the lump sum was seen as a loan that the state would pay back, but most of the time the state just took the lump-sum and moved on.

Now, this is an auction process – various people are bidding for the rights to tax and the government takes the highest bid. So not only does the winner needs to collect a sum equal to what they paid for the license (in order to break even), and then has to keep on taxing above that to turn a profit, but because there’s competition, their initial bids could quite possibly exceed the normal amount of revenue that a province could produce, let alone the added burden of the farmer’s profit margins. 

This is somewhat useful for relatively weak states – they don’t have to pay a large bureaucracy of tax assessors, collectors, and accountants to keep track of the money. Lump-sum payments means that you have more cash on hand at certain periods of time then if you were waiting for money to come in from different regions at different times of the year – and that’s very useful when you’re starting a military campaign and need supplies and a paychest to keep the army going. 

But, it often means higher tax rates than if the state was collecting it themselves, which increases discontent in the provinces potentially leading to revolts that you then have to put down with your army, and reduces the growth rate of the provincial economy because money is going to tax collectors (both as payments and as investment flows into buying up those contracts) rather than being invested in agriculture or manufacturing. And in the long run, it’s better to have a growing economy than a stagnant one. 

So….to answer your actual question: how did it work? 

• Tax farmers both employed their own leg-breakers debt collectors and had the right to call in royal troops. This is one of the things that made the practice less useful for governments than was originally thought – you’re supposed to be saving money on tax-collecting bureaucracy, but now you are detaching soldiers to protect tax collectors. And that means that your soldiers, rather than protecting your provincials from bandits and raids from enemy kingdoms, are shaking them down for money. That’s not good for loyalty to the regime. 
• In terms of the rates, it depends on the place and time. In the Roman Empire, tax rates tended to be set at whatever rate the auction price plus profit hit, but it’s also true that taxes were basically assessed collectively for certain cities and regions and local elites figured out the process of how to gather the money, usually by paying the tax collector out of their own pockets and then repaying themselves by taxing the lower orders. In medieval England, however, feudal grants given in “fee-farm” clashed with the fact that rents tended to be fixed sums established by written contracts or by custom.
• Tax season in premodern eras tended to be centered around the agricultural season, because after the harvest is when 90%+ of the population had something worth taxing and had it all in one place where you can easily count it and grab your cut. 
• In terms of how people felt about it – well, peasants tended to hate it, because they were the ones paying most of the taxes, and while they generally accepted the legitimacy of the King or Emperor, that got really fuzzy when some pushy businessman is waving some piece of paper at you that says he has the right to seize your property. Likewise, because the agricultural cycle doesn’t necessarily synch up with the auction process back at the capitol, a bad season could mean that you’re paying the old rates or higher with much lower incomes. Merchants tended to be of two minds – to begin with, they’re usually the folks engaged in tax farming, so those guys are very pro-tax farming. However, when it comes to contracts to collect tolls or excise taxes, merchants were the ones paying those taxes and weren’t happy about it. Unless the nobles were the ones letting out the contracts, nobles tended not to like it – tax collectors are upjumped commoners coming onto your estates and pushing around your people, and that’s your job. And when the pushy tax collectors caused a riot or whatever, you were the one they would call in to provide protection. And if the whole thing started a rebellion, you would get the blame from the central government.