First of all, you're a bit mistaken as far as the Red Baron goes. The Red Baron scored only 19 of his victories in the Fokker Dr.I triplane - the bulk of his victories were in more conventional biplanes, particularly the Albatros D.III. Manfred von Richtofen only adopted the Dr.I in late July 1917. The problems and tradeoffs of triplanes meant that not all pilots made the decision to shift from their biplane fighters to triplanes, and air forces on both sides of the front would ultimately shift back to biplanes and monoplanes by the end of the war.

​

The biplane craze started on the British side, with the appearance of the Sopwith Triplane on the front in early 1917. The Sopwith Triplane offered a few niche advantages owing to its shorter wingspan and larger wing area. Wing loading was generally lower than other fighters, allowing for an impressive climb rate, higher service ceiling, and good horizontal agility, and the narrower wingspan allowed for quicker rolls. The markedly different flight characteristics of the Sopwith Triplane were a shock when it first appeared on the front, sparking a wave of Triplane development in Germany that ultimately led to the creation of the Fokker Dr.I.

Fokker's triplane took on largely the same advantages of the Sopwith - it had a good climb rate and horizontal maneuverability. Unfortunately, as German pilots quickly discovered, it also shared the chief drawbacks of triplanes. Larger wing area and more supporting material between wings led to higher drag that left the triplanes slower than their biplane contemporaries, particularly in a dive. Perhaps most critical, however, was the danger during takeoff and landing. Like many fighters of the time, the Fokker and Sopwith triplanes employed rotary engines. A rotary engine rotated with the propeller - a feature that helped keep it cool during flight, but created dangerous flight characteristics. A ~400lb engine spun at several hundred RPM at the front of an aircraft with a loaded weight of ~1,500 lbs creates problems with momentum that make maneuvering especially dangerous. Triplanes gained a reputation for being particularly tricky to land, as the narrower wingspan created a lower moment of inertia that made the aircraft particularly susceptible to the gyroscopic effects of the engine at low speeds.

Despite the deficiencies, the Dr.I did remain in service into 1918 and it would continue to be flown by other famous pilots such as Lothar von Richtofen. But future fighter development was focused back on improved biplanes like the Fokker D.VII, and beyond the first generation of triplane fighters, there wasn't really any significant consideration or development of a follow-on triplane.