This is what I was able to come up with:
In 1795 Murdoch developed a replacement for Isinglass, a precipitate made from sturgeon used in the clarifying of beer to remove impurities, which had to be imported from Russia at great expense. Murdoch's replacement was made from dried Cod and was much cheaper than the 25 shillings a pound which Isinglass cost. This cost saving was so attractive that the Committee of London Brewers paid £2000 for the right to use his invention.
Murdoch's Isinglass replacement was so effective that in a court case brought by the British Customs and Excise Authorities, the noted Chemist, Sir Humphry Davy in answer to a question on whether it was "proper to be used for the purpose of fineing beer" testified that:
I believe it is if properly prepared - it is the same substance as Isinglass.
Use of Murdoch's "Isinglass made of British fish" continued and played an important role in reducing British brewers reliance on imported raw materials
Isinglass was originally a glue:
Glue can be extracted from fish by heating the skin or bones in water. The purest form of fish glue, made from the membrane of the air bladder (swim bladder) of certain species of fish such as the sturgeon, is also called isinglass (fig. 1). Isinglass can be produced from various species of fish using diverse manufacturing processes. Depending on the manufacture, the purity of isinglass can vary. Historic sources do not always specify which part of the fish was used to make the glue.
There is no record telling us exactly when and where the first animal or fish glue adhesives were used. However, it is known that at least 3500 years ago these adhesives were used in Egypt. Even though Egyptian records do not describe in detail the adhesive preparation process, they do tell us that it was made by being melted over fire and then applied with a brush (Darrow 1930, 9).
From the first-century Roman scholar Plinius we learn that two kinds of glue were used in antiquity: animal glue (taurokolla in Greek, gluten taurinum in Latin), made from the skins of bulls, and fish glue (ichtyokolla) made from some parts of fishes. In references to the glue used by ancient craftsmen, both terms xylokolla (in Greek) and gluten fabrile (in Latin) are cited; however, it is not clear to which kind of glue these terms applied (Gug 1975, 37).
In an eighth-century European manuscript from the Cathedral of Lucca, fish glue is recorded as a material for painting. A. P. Laurie translated this manuscript into English in 1926; it tells that the pigments in fresco paintings were applied to wet plaster without mixing them with a binding medium, using only water. For panel paintings wax was mixed with the pigments, and for illuminating parchment manuscripts fish skin glue was used (Laurie 1926, 107).
In the Middle Ages in a twelfth-century treatise on methods and recipes for painting and illuminating by the German Benedictine monk Theophilus, fish glue appears once again. In his Schedula Diversarum Artium (Ch. XXX) he gives directions for grinding gold and then mixing it with fish glue for use in gilding of illuminated manuscripts.
I suppose that someone just came up with the idea to put glue in the beer to see if it would stick the yeast together and drop it out. It worked and so it's use as a fining agent was established.