Like many things in history and life: it was not one thing but a confluence of factors. In no particular order:

First, it was on the western side of Scotland. As trade with America increased in the 1700s and 1800s, the importance of Glasgow as a shipping center increased. After the Act of Union in 1707, Scotland and England became part of the UK and were, effectively, like a free trade zone. Imports from English colonies in America could be delivered to Scotland; and Glasgow was right on the path. This isn't directly relevant to why so many ships were built there around 1900; but it would have never become a major shipyard if it wasn't an important center of trade with docks, which started 200 years earlier.

Second, the Scots had been working on deepening the Clyde throughout the 1800s. First with jetties, wing dams, etc to narrow and deepen the channel, and later with dredging. This made it better for transit, and would later become important for shipbuilding.

Third, most Scottish coal mines were in a geographic belt running east to west across Scotland, roughly across the narrowest part of the country. This belt crossed over the Clyde, and runs between Glasgow (which the Clyde runs through) and Edinburgh. Coal was needed for industrial production (directly to make steel, indirectly to power everything else involved). In earlier eras (Tudor era, Elizabethan, 1600s, etc.), most English shipbuilding was in the south, and relatively close to London. Ships were built from wood, and Scotland was either a foreign country or an area prone to rebellion. As shipbuilding transitioned to the use of iron and steel, and because Scotland was sufficiently integrated into the U.K., shipbuilding migrated away from southern England and into northern England and Scotland.

Fourth, Scotland was part of the UK, at the height of its maritime empire. The “sun never set” on the British Empire and that required a lot of merchant and war ships to connect all those locations. As the empire expanded, and as ships transitioned to steel construction, it was good economics to build the ships close to the coal and steel production.

Those factors all contributed to the rise of the Clyde as a major shipbuilding area.

It declined mostly because of WWII. Directly, it was heavily bombed by the Germans during WWII. Indirectly, Britain remained economically crippled after the war, the Empire and overseas possessions shrank, and Asian shipyards began building ships cheaper.