While watching a Korean drama called Gwanggaeto, the Great Conqueror and seeing how the king is praised by Koreans, I am curious to know what historical factors led to Kokuryŏ being able to conquer its neighbors
One must consider both the strength of Goguryeo and the weakness of its surrounding states.
Strength of Goguryeo
Goguryeo had seen severe military setbacks during the reign of Gogugwon at the hands of the Earlier Yan polity and Baekje— Gogugwon was killed in battle by Geungusu, crown prince of the latter. Geungusu failed to follow up on the victory and proceed with the conquest of Goguryeo, allowing the latter to regain its strength.
King Sosurim of Goguryeo enacted a number of reforms to strengthen royal authority and unify the state. Buddhism was encouraged to solidify national spirituality, and Confucian academies trained bureaucrats to serve the royal administration. He also promulgated a code of laws.
Weakness of Goguryeo's neighbors
On Goguryeo's northern borders, there was turmoil as the Earlier Yan kingdom collapsed and gave way to the Later Yan. After the reign of Murong Chui, the Later Yan was distracted by a succession of palace intrigues and territorial pressures from the Northern Wei dynasty and the Jin dynasty.
To the south, the third of Korea's "Three Kingdoms," Silla, was becoming organized a centralized kingdom. They established relations with Goguryeo for defense against Baekje, which had allied with Wa (Yamato Japan) and invaded in 399. The King of Silla offered himself as a vassal in exchange for support, which came the following year.
Success in battle
Thus, the stage was set for an ambitious king like Gwanggaeto to expand territory greatly. Jinsa and Asin of Baekje were weak warriors, and their mass conscriptions alienated their subjects and caused some to flee to Silla. After steadily losing territory, Asin prepared his capital at Wiryeseong against a land invasion in 396. Gwanggaeto attacked from the river instead, catching them unprepared, and subjugated Baekje. In 399 as noted above they rebelled with the assistance of Wa, but were defeated.
With Baekje and Silla neutralized in the south, Gwanggaeto could turn his attention to the north. The Later Yan, weakened by misrule and warfare with the Northern Wei, were expelled from the Liaodong Peninsula, and their successor, the Northern Yan, made peace in 408 and survived as a vassal. That in turn freed Gwanggaeto to annex the remnant Buyeo in 410.