Specific examples of timocracy would be early Athens (7th/6th century) and the Roman Republic, both of which restricted political and particularly military offices to particular classes, determined almost entirely by wealth in the former, and by a combo of wealth and social status in the later. Timocracy itself is alternatively derided and lauded depending on which ancient philosopher you are reading. That said, an 'absolute' form of timocracy never really existed as it is described in works like Plato or Aristotle, and neither are entirely consistent in their definitions thereof.

As for how it worked, in practice it would be pretty simple. You divide the polity up into classes, say four or five, and then give each class a certain set of rights and responsibilities. The motivations for this in the ancient world seem almost entirely on the face about military responsibilities, which then earn rights related to those responsibilities. Indeed, for most of these systems, it the necessities of equipping and marshalling armies that dictated the breakdown of the classes, and from there the distribution of political powers and rights flowed.

Let's use the Roman Republic as an example. Timocracy and Aristocracy are often conflated (and Plato himself describes Timocracy as a butchered Aristocracy). However, they are very closely related. Indeed, in Rome one could argue that both an aristocracy and timocracy existed. When I use aristocracy in this sense, I am referring to Plato's ideas for the record, i.e. 'rule of the best'. In this sense, Rome as a whole was broken down into two 'social' classes, the patricians and the plebians, with the patricians being the aristocrats. In theory. There were a lot of political battles over this. However, outside of this distinction, there were six property classes, ranging from the rich senators and equites (origin of the term knight) down to the proletarii, the very poor. And, while the lower four classes had political power, the levers of power were dominated by the wealthy. These classes were based on the required wealth to level equipment (for example, equites refers quite literally to people who could afford a horse), but from them flowed the access to control over the elected magistracies and the courts, the two main day-to-day methods of control in ancient Rome. The wealthy classes also possessed more power in the assemblies, where they were over-represented compared to the lower orders, and also controlled the very influential Senate of Rome, which although for most of the history of Rome had no de jure powers, had enough de facto power to set the course of the state.

I am not personally aware of any suggestion that Sparta operated on a deliberate timocratic system; rather, it is a critique of later Spartan excesses in the 5th and 4th centuries BCE rather than an explicit design of the system, and related to the growing concentration of property there.