Simply put, Russian banks (and other institutions) had local assets and hard-currency liabilities.
Local assets lost value not only because they were denominated in a currency that unexpectedly depreciated by a lot, but also the equities market crashed, and a lot of the corporate debt defaulted following the government default (local-currency debt).
Hard-currency liabilities were: hard currency deposits at the banks. Those Russians who held their savings in hard currency at banks that did not collapse (e.g. at foreign banks) got a windfall in terms of the local currency. Hard-currency bonds - Russian corporates (including the banks) and municipals (various cities and oblasts) were very fond of issuing eurobonds.
Furthermore, the statement that August 1998 events were caused by anti-inflation measures sounds very questionable to me. While there was a lot going on that summer, the immediate reason for the sovereign default was:
The government (which was at least as corrupt and kleptocratic as the current one) was issuing local-currency debt (GKO - государственные краткосрочные облигации) that the general public was not allowed to buy. The could only be bought by the government officials, their cronies, and foreign investors who bribed the officials in order to be allowed to buy GKOs. The yield on the GKOs (around 50% a year in July) was much higher than the rate of local currency inflation or the returns one could get on equities or corporate debt. The government kept selling more GKOs in order to pay off the previous bond holders, and expanded the list of people allowed to buy GKOs from a few "special" friends to a much wider audience.
While it was clear to everyone that the GKO pyramid scheme was not sustainable, there were two views widely held by emerging markets investors.
Some people felt that this was too weird, and mandated Russia exposure to be flattened during the run-up to the default.
Many had bet that Russia would eventually print local currency to pay off the GKOs and stop issuing new ones. This would inflate the currency. This scheme was carried out by many emerging markets governments in the 1980 and 1990s.
(I was "lucky" to work at a firm where two parts of the firm took opposing views on Russia at the time, which was fun to watch.)
In August, Russian government decided to default on the GKOs. It is extremely unusual for a sovereign to default on its debt denominated in local fiat currency when instead it could just print more local fiat currency. In fact, I only know two times in history that it happened: Russia in 1998, and Peru during Garcia's first term.
Strangely, Russian government did not default on its external-law hard-currency debts (e.g. eurobonds).
Many Russian corporates (including banks) and municipals decided that the GKO default was a good excuse to default on their own obligations - local and external law, local and hard currency, even if they were solvent.
Following the avalanche of defaults, local currency devalued - probably much more than it would have if the Russian government just printed local currency to pay off its local currency debts, as many other emerging markets debtors had done in similar cirumstances.
If your source refers to the Russian government defaulting on its GKO's as "anti-inflation measures" then either they don't know what went on (the most charitable explanation), or they do know, but they have some agenda, and intend to mislead the reader.