Essentially with the rise of the German research university model. Humboldt University of Berlin was founded in 1810 and while a good portion of its curricula was still founded on the classics, law, philosophy, medicine, and theology, the sciences all of a sudden had a place at the table. In the United States, schools like RPI and Johns Hopkins were established based on this similar model. At the same time, President Charles Eliot steered Harvard to a more research oriented university. Why was this happening, though?

For centuries prior, there had been growing resistance to the dominant mode of scholasticism, a type of education built on dialectical reasoning based on Classical Greek and Roman thinkers and works. Peter Ramus in the 16th century was perhaps the most vigorous critic of scholasticism. He was actually thrown out of the university of Paris as a graduate student for criticizing scholasticism and his dismissive attitude to classical thinkers before being readmitted and finishing once a friendlier administration came to power. Thinkers like Francis Bacon were trumpeting "The New Instrument" (science), a play on Aristotle's Organon (his method of knowledge production).

Things came a head during the 18th century Enlightenment. Like many sectors of Europe, education was at a crisis. With Christianity being unmoored as the transcendent anchor upon which knowledge was established, Western thought needed a new base. Science was beginning to demonstrate its potential. Advanced math, calculus, blood circulation and new medicine, breakthroughs in optics and physics, and the beginnings of chemistry were astonishing. Biology and plate tectonics were also in their infancy. Darwin was just around the corner.

Ultimately, education isn't a part of a vacuum. Society and culture changed which caused education to change, and education changed which caused society and culture to change.

Previous notions of the educational system relied on either learning skills to be employed and/or a grounding in an education that would demonstrate your ability to be a part of the ruling class (knowing French, Latin, the Bible, Plato, etc.). With the rise of research oriented universities, creating new knowledge became a more fashionable goal of education. This is not to say the other educational modes have died out, but it's a reason why learning the classics has largely died out.