It just seems like that region use to be very prominent with the Silk Road and the Mongolian Empire, but now I barely hear about it. Why is this?
Well, there are certainly quite a lot of factors, but I would like to break them down into three major reasons:
1 Decline of the Mongol Empire and the Silk Road
The two things you said, the Silk Road and the Mongolian Empire, were the two things that put Central Asia at the forefront of Geopolitical significance.
To provide context, before the Industrial Revolution that made the European powers rich, China was the industrial powerhouse of the Middle Ages up until the 16th Century. /u/cee2027 has a fabulous explanation regarding China's economic reach here
In between the manufacturing capital of the world and Europe was of course Central Asia. The Silk Road and the important trading cities along it made Central Asia pretty rich and important too. Almatu/Almaty, Rey, Samarkand, Merv and all the other cities flourished because of the Silk Road Trade. When the Mongols came in and instituted a pax mongolica trade flourished even more.
However, two key things happened that disrupted the importance of Central Asia as a bridge between Europe and China. First was the civil wars that broke the Mongol Empire. With constant wars and the breakdown of the royal roads, trade died down as the central government that protected and maintained the roads collapsed. The second event was the Black Death, which did not only kill off quite a lot of people, it killed off quite a lot of people wanting to move, buy, make and sell goods.
2 The Ottoman Empire
Another reason why the Silk road died down in significance was that the Ottoman Empire happened. When the Ottomans successfully captured Constantinople in 1453 and completed the conquests of Arabia and the Mamlukean Egypt in 1517, they had sole control over the trade routes of spice from Asia. Europe was especially concerned about this, since this meant facing higher prices, instead went and found new routes to India and China, with the Portuguese starting with the efforts of King Dinis of Portugal and of course with Henry the Navigator.
3 The Age of Discover and the Industrial Revolution
These are the final nails to the coffin of Central Asia as being in the forefront of geopolitical significance (at least for now). With Europe having maritime trade routes replacing what the Silk road did in the past, there was not much reason for Central Asia to have much geopolitical significance as the bridge between China and Europe. It had a spot in the limelight during the Great Game if the 1830s, but aside from that, geopolitics has dictated that focus of the world's attention would instead move elsewhere.
SOURCES: