The short answer: guns.

In particular, the breech-loading rifle makes the mounted archer thoroughly ineffective. The breech-loading rifle didn't bring about the end of cavalry, but it did force changes in cavalry tactics and training (Phillips 2011). Part of that was that cavalry standing at a distance and exchanging fire (whether musketry or arrows) with infantry was no longer viable.

Before that, effective long range missile weapons could make mounted archery ineffective. For example, ancient Chinese massed crossbowmen could be very effective for defence against mounted archers, and in later times, musketeers. The lower rate of fire of crossbows and muzzle-loading guns compared to the mounted archer's bow meant that the mounted archer was not yet useless. Given armour, the effect of crossbows against the mounted archers could be reduced, and armoured mounted archers continued to be effective until guns became plentiful enough. With the advent of the breech-loading rifle, the infantryman could fire accurately and often, and reload behind cover, and shoot through any armour the mounted archers wore. Add long-range artillery to the mix, and one has a battlefield where cavalry cannot stay in effective range of infantry and artillery and survive for long.

This wasn't a sudden thing. The growth in the number of effective long-range infantry missile weapons (initially crossbows, and later guns) eroded the effectiveness of mounted archers. The 17th century growth in the proportion of infantry armed with guns (i.e., musketeers) was a significant blow to mounted archers. The difficulty that infantry had in actually catching mounted archers (or cavalry of any kind) meant that the bow could still be a useful cavalry weapon, and the bow remained in use long after the musket was in common use.

The advantages of the gun over the bow (range, damage, penetration of armour) meant that the gun could be an attractive alternative to the bow for cavalry. The difficulty of loading a muzzle-loading long arm on horseback resulted in a number of possible choices: (a) just use the bow, (b) use the pistol, which can be loaded on horseback (but is inferior in performance to long-barreled firearms), or (c) carry a musket and a bow - use the musket first, and then switch to the bow. Choice (c) was common in Central/Inner Asia, seeing common use in Tibet and Mongolia.

The development of breech-loading rifles offered cavalry a greatly superior alternative to the bow, the musket, and the pistol. At this point, the horse archer is obsolete. This obsolescence was driven from both ends - infantry firepower growing to the level where cavalry need to exploit cover, and close the distance to the infantry as quickly as possible (Phillips 2011), rather than staying at a distance and shooting arrows, and the development of greatly superior alternatives to the bow. Mounted archers remained in use where insufficient modern guns were available, but their performance in battle against modern-equipped forces was often poor (e.g., traditionally-armed Qing (Chinese) cavalry in the Boxer War in 1900). Cavalry - equipped with modern rifles - stayed in effective use until after World War 1, and continued to perform some of the military functions of the mounted archer after the mounted archer was gone.

Similar changes in the value of mounted archers took place in Eurasia and in North America, as muzzle-loading guns and, later, breech-loading guns became more common. There is a good coverage of the North American case in Bohr (2014).

References:

Bohr, Roland. Gifts from the Thunder Beings: Indigenous Archery and European Firearms in the Northern Plains and Central Subarctic, 1670-1870, UNP - Nebraska, 2014.

Phillips, G. (2011). ‘Who Shall Say That the Days of Cavalry Are Over?’ The Revival of the Mounted Arm in Europe, 1853-1914. War in History, 18(1), 5–32. https://doi.org/10.1177/0968344510382606