In pure practical terms, in Republican Florence the major professional associations, commonly referred to as the Arti Maggiori (roughly translatable as "Major Guilds") were the primary writers of laws, rules, and regulations. Each Arte Maggiore elected a Priore who sat in a council called Priorato delle Arti, more colloquially referred to as Signoria, which was Florence's powerful executive body. While the leaders of the Arti associations might draw an income from owning and managing businesses instead of actually working in a trade, or be granted an income from relatives active in a trade, the credit lines they drew from the Medici bank were vital (much like credit facilities granted by banks to modern businesses to cover operating costs). Of course, the Medici were only one bank of many, but being the largest and wealthiest meant that even when prominent dynasty members were exiled from the city by the Balìa (a powerful ad-hoc committee of prominent politicians) it was only a matter of time until the Priorato's rotation meant sympathizers (and debtors) were in the majority and reversed the sentence. Additionally, the Medici's ample use of "gifts," expenditure on public works, and patronage of the arts also worked to enhance their prestige and endear them to the leaders of the Arti.
How this happened is complicated. Within the Italian political ecosystem between the eleventh and fifteenth centuries, Florence is a strange place. On the surface, its political characteristics resemble those of its neighbors in Tuscany, its peers in the rest of Italy, and indeed many other urban tracts of the Mediterranean: vaguely republican in governance, certainly oligarchic and unequal in enfranchisement, and religious power and protocol is intimately entwined with secular authority. But Florence's institutional reality had some key differences with its peers.
Where Florence immediately stands out is the longevity of its "Republican" institutions. Indeed, Florence retains the pretext of republicanism and collegiate rule until the early 16th century. This is not common: when we begin to feel the political influence of the Medici sometime in the 1300s, just over the Appenines two political dynasties are locked in a destructive conflict with little to show for collegiate rule. The Visconti of Milan and Scaligeri of Verona had long dismantled the "communal" power structures in their respective cities in order to govern as despots; having consecrated control over their communities they could challenge each other for regional hegemony outside their city walls. Of course, the vestiges of collegiate governance still existed here and there, especially in communities farther from the centers of autocratic power in Milan and Verona, while in countless other communities (just think of Bologna, Genoa, or Florence's own neighbors Pisa and Siena) the "political class" voluntarily associated themselves with these autocrats (and, when possible and advantageous, extracted themselves from this sphere). Instead in Florence on the one hand there is the emergence of a dominant political dynasty as there is in other prominent Italian cities (our Medici), but on the other hand that dynasty permits the communal institutions to continue existing on the forefront as nowhere else.
So it is an anomaly that that the Florentine Republic continued to exist under Medicean rule, creating a "crypto-government" (a popular phrase in Italian scholarship). And that has everything to do with the method of the Medici's rise to power. While in other Italian cities internal political crises that the communal institutions proved unable to solve had catalyzed the emergence of autocrats, the Medici's influence in Florence emerged slowly and laboriously. It was certainly a rise that could also be wrought with crisis and conflict, but these were never conflicts that questioned the basic functioning of the political system as they did elsewhere, only calling into contention the Medici's place in that system.
The Medici were a populous dynasty divided amongst several branches. They are first documented in Florentine records in the twelfth century, but it's almost certain they arrived in the city much earlier. Their family business activity started in the wool trade, but would soon morph into the universal investing of the moneyed classes, eventually turning to banking by way of foreign exchange speculation. While several members of the Medici dynasty would hold public office between the twelfth and fourteenth centuries, the family member who lay the groundwork for their definitive control of the city was surprisingly indifferent to politics: Vieri di Chiarissimo Medici. The Chiarissimi branch of the Medici had cemented themselves in the foreign currency exchange business such that they were sometimes just referred to as the "Medici di Cambio" or just "di Cambio" (meaning the "Medici Exchangers" or just "the Exchangers"). While Vieri di Cambio made a name for himself as an aggressive foreign exchange speculator, in 1348 he moved into aggressive, if not downright predatory, lending. While Vieri's relatives embroiled themselves in local politics, Vieri himself remained focus on his business. But politics would soon come to Vieri in spite of his disinterest: after Florence was subject to a period of civil strife in 1378, Vieri was acclaimed to public office. While his business practices were characterized by a high risk-reward ratio, in politics Vieri had so few enemies that he was kept in power even after the populist government that had called him to service collapsed.
Vieri retired in 1393, and his sons were unable to keep his financial empire solvent. But Vieri's most trusted employee, his nephew Giovanni of the Bicci branch of the Medici, had bought out and spun off his uncle's Roman office and had managed to secure a portion of the Vatican's revenue to repay gold he had fronted to various Popes and Anti-Popes. Like his mentor Vieri, Giovanni's might have taken risks in his business affairs but distanced himself from politics. Indeed, Giovanni continued to be disinterested in political affairs after he moved his seat of operations back to his native Florence in 1397, but also like his uncle Vieri, Giovanni found himself tapped for extremely important positions within Florence's government in spite of his disinterest (in fact, his "official" political career would far surpass any other Medici). Giovanni quietly used these elevations to grow his business interests further, and by the time his son Cosimo took the reins of the family enterprise, the Medici were in a position to control Florence both financially and politically even without holding public office.
Through a combination of personal and business connections, Cosimo was owed money by almost every single guild leader and could influence their members accordingly. Cosimo's status as Florence's richest man also made his council and advice much sought-after, creating a strong social network of Florentine elite recognizing him as leader. Under Cosimo's successors, the business-focused aspect of control probably declined a bit, but was replaced by more lavish gift-giving coupled with stronger social ties. Of course, this control was always precarious, easy to challenge, and ultimately came very close to being undone in the early 16th century. But that's a different topic.