Garum production and consumption declined because increasing instability in the late Western Empire disrupted production and trade (and not just of garum). Further, a key ingredient - salt - became too expensive and pirates undermined the trade routes. Note also that garum was far from the only thing that declined or disappeared along with the empire; so too did many other features of everyday Roman life. However, it could still be found centuries after the fall of the Western Empire, especially in the Eastern Mediterranean.
According to the Italian archaeologist Claudio Giardino of the University of Salento (Lecce),
"In the Roman times, salt was a cheap material....When the Roman
Empire collapsed, they put taxes on the salt. And because of these
taxes, it became difficult to produce garum."
Cited in 'Fish Sauce: An Ancient Roman Condiment Rises Again'
Also, as the Romans lost control of the Mediterranean, piracy increased:
"The pirates started destroying the cities and the industries nearby
the coast. You could be killed any moment by the pirates, without the
protection of the Romans,"
In order for garum to be produced profitably in large quantities but at prices which most people could afford,
...large fishing fleets were essential and so were beachfront
facilities (especially after regulations were put in place by
Constantine Harmenopoulus that garum works could not be within a
certain distance of a town due to the odors) and safe shipping routes
from areas of production to faraway clients. Other factors that may
have affected garum pricing and access were the requirement of a
large workforce, land for facilities, and credit during a tumultuous
time where shipping routes would not necessarily be secure. The
Empire’s decline and the contraction of the Empire pulled apart the
trade routes and threads from production to the client.
Source: T. Carpenter & G. Stern, 'The Rise, Fall, and Rise Again of Garum'. Paper presented at the Northern Great Plains History Conference, September 2016.
The location of garum production sites away from cities may have meant that, in some cases at least, they were less well-protected from raiders than they would have been had they been located inside walled cities. With this "loss of order", it was not just garum consumption that declined with the decline of the empire; so too did other industries, as well as other features of Roman life, like the construction of roads and
making a daily visit to the public baths or wearing the toga.
Source: Carpenter & Stern
The spread of the popularity of garum to provinces such as Gaul had occurred with the Romanization of these provinces when Italians settled there:
Augustus settled many Italians in the provinces in the settlements
of the 20s BC. One mostly overlooked way they Romanized the
provinces, besides recreating grid-pattern Roman cities with baths and
a forum, was to bring their tongues with them, meaning both the
speaking of Latin – on which much has been said and written before –
but also the taste for Roman food. They imported and eventually
produced their own garum to enjoy the flavors of home far from Italy.
As the Romans withdrew, so declined the popularity of garum and the expertise to make it. The fact that it did not immediately disappear is an indication of a degree of Romanization among the indigenous population.
Concerning the decline of the trade and consumption of high-priced garum, another factor to consider is the general decline in wealth in mainland Italy especially. The high-priced variety often came from afar as
Some distantly-made garum was a status symbol (like Belgian beer or
Source: Carpenter & Stern
However, it did not completely disappear with the fall of the Western empire in some regions. For example,
...it remained in a few little pockets — like in Southwest Italy,
where they produce colatura di alici, a modern descendant of the
ancient fish sauce. The product was barely known even in Italy just a
few years ago, but it is gradually being rediscovered.
Also, manufacture and consumption continued for a while in Tunisia and Gaul (despite the physician Anthimus writing it should be 'banned from every culinary role'), and in the Eastern Mediterranean as late as the 16th century.
There is some controversy as whether Asian garum (the production of which " has been uninterrupted for centuries") was a 'technology transfer' from the west as the origins of the product in Asia are somewhat obscure. Without lending any support to to the 'West-to-East' theory, Carpenter & Stern note that
Roman and Vietnamese fish sauce have a similar ratio of fish to salt ≤
5:1, but the Romans fermented garum for less time before bottling it.
Both Roman and Southeast Asian cuisines use fish sauce in similar
manners, as both an ingredient in cooking and as a condiment that
can be diluted with other ingredients like vinegar or sweetener.
Fish sauce remains popular in Asia and is now also found in West African cuisine (where I first encountered it).