It has to do with the fact that the war had delegitimized the Joseon central bureacuracy, as well as the personal weakness of the king, Seonjo in his reign.

As you may know, the Joseon army, the army of the state, was completely and utterly destroyed in the opening months of the Imjin War. In the battle of Tangeumdae, the Joseon Army was so soundly defeated that in only 20 days, the entire court left Seoul (then called Hanyang), and made way to Uiju (now Sinuiju), on the border between Korea and China. Thus, what we see is a rapid loss of authority, as the various regions were left for themselves to fight and resist.

And fight and resist they did- many literati class- the confucian scholars that were regional authorities in the countryside began to form "Righteous Armies" aka local militiamen to fight against the Japanese, outside of the central order of the state, and while the court was glad that there were armies fighting still, any armed force outside the control of the central government operating in the country is a potential threat.

These righteous armies as they were called were an important part of the Korean resistance to the advancing Japanese army. Yet, in the lull of the fighting between 1593 and 1597, there was a revolt led by one of these "Righteous Armies" operating in the countryside, called Yi Mong Hak's Rebellion in 1596. Yi, a man of the royal bloodline led a revolt, calling for the end of the incompetent King who had abandoned the people. While this revolt ends in failure, this event marked the end of the Joseon court's patience with the "Righteous Armies" and many prominent leaders of these militia (there was no ONE militia operating, but rather these were almost warlords defending their hometown, with not much rigid organization between them.), including Kim Deok Ryung (김덕령) were arrested and tortured for their alleged conspiracies.

So how does this connect to Yi Sun Shin?

Interestingly there are claims that it is around this time that Yi fell out of favor with Seonjo, as to any monarch, a general winning battles while the king ran away, with an army capable of defeating the army of the king, who was virtually the regional governor of the breadbasket of the country, the only part of the country that had not yet been ravaged by war is a huge threat. Thus, it is around 1596 that Seonjo began to favor Won Kyun over Yi- before the war, if you look at Yi's career, Yi was promoted multiple times within a span of a year, a sign that he was considered as one of the biggest "prospects" if you will.

The Annals of the JOseon Dynasty, the official court history, states that Yi was promoted from a 현령, a town mayor/governor, to the Left Commander of the Jeolla Navy-

선조 25권, 24년(1591 신묘 / 명 만력(萬曆) 19년) 2월 16일(계미) 2번째기사 (1591 Feb 16th,) 사간원이 전라 좌수사에 초수된 이순신의 체차를 청하다. 사간원이 아뢰기를, "전라 좌수사 이순신(李舜臣)은 현감으로서 아직 군수에 부임하지도 않았는데 좌수사에 초수(招授)하시니 그것이 인재가 모자란 탓이긴 하지만 관작의 남용이 이보다 심할 수 없습니다. 체차시키소서."

하니, (선조가) 답하기를,

"이순신의 일이 그러한 것은 나도 안다. 다만 지금은 상규에 구애될 수 없다. 인재가 모자라 그렇게 하게 하지 않을 수 없었다. 그 사람(이순신)이면 충분히 감당할 터이니 관작의 고하를 따질 필요가 없다. 다시 논하여 그의 마음을 동요시키지 말라." 하였다.

The Officer of the Censor stated, "Yi, as a mayor/governor (of a small province (현 "hyun" was the lowest level of governance in Korea, something equivalent to a small village) had not even reached his post, yet he was just promoted to the Left Commander- while there is a shortage of good men, this is a misuse of our public offices"

The King replied, "I understand that the manner of Yi's appointment is as thus. However, right now we cannot be held to protocols-we are in a shortage of able men, so I had to do as thus. He (Yi Sun Shin) is capable enough to do his job, even if his rank has been low, so do not speak of this matter again and shake his mind"

Yet, by 1597, Yi is taken and arrested, nominally for sending a falsified account of the successful burning of the Japanese encampment in Busan. Even today whether he was guilty or not is debated, but this was a great chance for Seonjo, and thus Yi gets tortured, and sentenced to death, until the great defeat at Chilchonryang decimated the Joseon Navy.

In short, it was the circumstance that the central government, the object of Confucian ideal of loyalty, lost its authority in the Imjin War as the state army had been soundly defeated and the King ran away from Seoul. Thus, the King actually had to fear for his throne, and a general that had not lost a major battle up to that point, the only one winning victories left and right that governs a big chunk of the Kingdom and has a sizeable force would be a threat to Seonjo's throne. You have to remember at this point, in the point of view of Seonjo, his kingship was the country itself.