My experience lies in the early modern period, not the middle ages, but much of the logic underlying the phenomenon of tax farming applies equally to both periods. Since we're talking about a broad phenomenon across a large span of time and a sprawling geography, I'm going to be generalizing a bit in this answer.

Tax farming is the sale or lease of the right to collect a given source of revenue to a private person (or persons). This was an extremely widespread phenomenon in world history. The advantages and disadvantages associated with it would vary depending on the specific circumstances in each case of its implementation, but I would identify three primary reasons why it was so attractive. First, because it made irregular and unpredictable revenues regular and predictable. Second, because it transformed in-kind taxes into ready cash. Third, because it could be used as a mechanism to anticipate future revenues.

Medieval and early modern monarchs were generally unable to predict with certainly how much the taxes owed to them would amount to and according to what schedule they would be collected and remitted to them. In some cases this was a built-in feature of the taxes themselves: income from something like customs revenues depends on the amount of goods being moved by merchants, which varies heavily throughout the year and according to ever-changing economic conditions. Even theoretically more stable revenues like land taxes would vary depending on the climate and quality of the harvest. Not only collection from the taxpayers varied, but also the timing of the tax collectors' remissions to the monarch's treasury could be unpredictable. The monarch who establishes his own tax-collection administration would have to live with these uncertainties as well as shouldering the financial burden of creating such an administration in the first place. By selling or leasing the right to collect to a tax farmer, the monarch could place the burden of creating an administration on the farmer, and in the lease agreement could specify ahead of time just how much money he expected to receive, as well as the schedule according to which he expected to receive it. The messy business of actually getting that money would fall to the tax farmer, while the monarch could go about planning his expenses on the basis of the regular income he expected to receive in return (assuming that the tax farmer wouldn't fail in his obligations or go bankrupt).

In the medieval and early modern periods, many taxes were paid not in cash, but in kind. Cash money was generally scarce, but monarchs needed cash in order to meet their obligations, above all to pay for soldiers. Selling the right to collect an in-kind tax would allow the monarch to collect usable cash rather than face the difficulty of trying to transform the in-kind tax into something useful.

Lastly, tax farming could at least in some contexts be used for revenue anticipation. More commonly, tax farmers were leased the right to collect a tax in exchange for paying a predetermined amount of money to the monarch according to a fixed schedule. But if the monarch demanded a significant amount of this money up front, this would allow him to meet current expenses on the basis of revenues that hadn't been collected yet. In particularly dire times, monarchs could sell the right to collect certain taxes months or years in advance, bringing in immediate cash while undermining long-term income. This can also be understood as a form of debt, and granting the right to collect to the creditor was the means of repayment.