Different state-owned industries were privatized differently, mainly based on their size. Smaller businesses were sold to local governments who could then decide what to do with them. The medium- and large-sized businesses were given options how to proceed with privatization, often choosing to either give the workers 25% of the business for free (but making these shares non-voting) or selling 51% of it for a low price (and allowing these shares to vote).

What largely happened is everyone in the country was allowed to buy vouchers—small shares of a given industry—for cheap (or was given them for free) and almost everyone who could did buy them. But because everyone was given such an opportunity, individually these vouchers did not hold much ownership. After 1993, the vouchers were allowed to be sold and traded freely on the market, so individuals who had the money started buying the vouchers from people, amassing more and more control of an industry the more they bought.

Some of the oligarchs came from this transition because they were able to buy control of industries by buying many vouchers from people. Others came from having close ties to the government prior to privatization, having been managers for the state-run industries during the Soviet time, and were more or less given ownership when their business was privatized.

Sources that describe the process in much more detail:

https://scholar.harvard.edu/files/shleifer/files/privatizing_russia.pdf (Pages 148-152 especially)

https://hal.archives-ouvertes.fr/hal-01176837/file/gurievrachinsky.pdf (Pages 9-10)