It's successful because there was no better alternative. Before that you could basically use Chinese characters with similar sounds (horrendously inefficient and distorts meaning of the Chinese characters), or just write in Classical Chinese.
This system gave Korean its own writing system, and a good, simple-to-learn one at that.
The hangul script was also successful because it was a state-sponsored improvement over using the chinese characters for the Korean language. Sometimes it helps to have someone abruptly enforce adoption of a better method.
Not everyone supported it. Although King Sejong, who had it invented in the 1440s, was a big proponent, prominent Korean Confucian scholars opposed it on the basis that it would dilute the sway that Confucian/Chinese culture had on Korea and turn them into barbarians. The real reason was probably that by keeping the writing system so time-consuming to learn, they would be able to control the poor much beter.
By the way, there was an excellent South Korean historical TV drama that aired late last year called Tree With Deep Roots (https://en.wikipedia.org/wiki/Tree_With_Deep_Roots) that covered the creation of hangul by Sejong and the opposition of the entrenched nobility to its promulgation (links to 720p video and English subtitles are available if anyone's interested).
In fact, hangul was banned by a subsequent king after peasants (who were previously illiterate because of the years it took to become educated in Chinese characters) made signs in hangul and started protesting outside his palace. It wasn't until the modernizing Gabo Reforms of 1894-96 that vernacular Korean replaced Classical Chinese as the official written language of the state.
And even then, words of Sinitic origin (which makes up 60% of contemporary Korean vocabulary) continued to be written in Chinese characters, with hangul used only for native Korean words, similar to how Japanese is written to this day (although Japanese uses Chinese characters for many words of Japanese origin as well). It wasn't until the early 1990s that big Korean newspapers, such as the Chosun Ilbo, started dropping hanja from the majority of their writing (though it is still used in a few situations).
Interestingly, when the state tried to get Korean to adopt the new writing system in the 15th century, it failed miserably because of various political and ideological issues. Only later did Hangeul actually begin to gain popularity, and this happened organically.
This system gave Korean its own writing system, and a good, simple-to-learn one at that.