There are several factors that leads to such success by Yi Sun Shin against the Japanese Navy.

1. Superior Ships - While "Superior" really is a subjective term, in terms of the naval battles, Koreans possessed superior ships for its tactics and environment. Koreans Ships, Panoksun, were flat bottomed ships that made it very stable, maneuverable and perfect for tidal plains of Korean coast. The ship would also have several levels, the bottom being rowers, and other levels being filled with cannons or archers, with the highest level towering over smaller Japanese ships. Pine, which was used as the material for building the ship was also very sturdy. The Japanese Ships were mostly much smaller and had V shaped bottom giving it much more speed, but little maneuverability. It's materials were Fir or Cedar, which made the ships structurally weaker than its Korean counterpart. Furthermore, because of its structural weakness, Japanese ships could carry considerably fewer cannons than the Korean ships. (Some people like to bring up Turtle Ships, but the last time i checked, the nature of Turtle ships were being debated, not to mention Panoksuns were much more integral to Korean navy than Turtle ships.)

2. Cannons were integral part of Panoksuns. With Korean ships being outnumbered, it fell to Korean ships having the advantage of cannons over their Japanese counterpart. Korean ships could carry considerably more cannons than the Japanese, A Korean ship could carry 26 cannons, while Japanese ships at best could only field 2-4 cannons per ship. Not to mention, Korean cannons were far superior to the Japanese cannons or arquebus with superior range and power. I can't recall what the range of these cannons are, but quick wiki search has told me that the farthest range Cheonja Chongtong is about 1.4Km. These Cannons would should rockets, tipped with metal, effectively disabling weakly structured Japanese ships from quite a distance. Not to mention, the Japanese naval tactic at this time was mostly trying to overwhelm enemy ships and board them, eventually overwhelming them in hand to hand combat. Obviously this tactic couldnt work if Korean ships were destroying the Japanese ships as they made their approach. This was one of the reason for the defeat at Chilcheonllyang, where Won Gyun, with the Korean fleet approached the Japanese fleet, and either did not use or did not have the chance to use the cannon from range, with the Korean fleet eventually being overwhelmed in hand to hand combat against the Japanese.

3. 1st Battle of Jinju - At the time of Japanese invasion, Yi Sun Shin's base of operation was in Jeolla province. Jeolla is known as one of the "breadbasket" of Korea at the time, and its capture would have allowed Japanese army to get their hands on the supplies. Jinju is important because it was one of the gateway in to Jeolla province from Gyeongsang province, which the Japanese had invaded. However, the defeat of the Japanese at Jinju, gave Yi Sun Shin freedom of operations, as Jeolla province was essentially secure.

4. Yi Sun Shin almost always chose the location of battle. Whether it was bringing the Japanese fleet to desired location or attacking, it was Yi who chose the battle ground. His essential tactic was to spread his ships out and bombard the Japanese ships from as far away as possible. This ranged bombardment was very effective against the Japanese tactic of trying to board enemy ships. Not to mention its superior cannons meant that Korean ships could out compete Japanese ships in ranged battles. Furthermore, Yi used tidal plains, and tides to his advantage. With flat bottom, Panoksuns were less effected by the movement of the tide, while V-shaped Japanese ships were constantly effected by the movement of the water. Furthermore, Yi utilized Crane Wing formation, which is essentially U-shaped formation that surrounded the Japanese ships and bombarded from all angles while the essence of the Japanese tactic was to try and move in, in order to grapple and board Korean ships. These kinds of traps would annihilate Japanese fleet.

It is true that Yi did not have formal Naval training and success in the army does not always transfer in to the navy, but it could simply be that he was a very careful planner and a very successful one at that. He did rise quite rapidly in ranks so that should somewhat account for his brilliance.