The easiest solution would simply be to sort the strings by length (using Java's built-in sort method) and then simply take the first three (or last three depending on the order in which you sorted) strings from the sorted array.
That solution would be much shorter, simpler and easier to understand than yours. However it's O(n log n) whereas your solution is O(n) - also if this is Homework, using sort might count as cheating.
To simplify your solution, the first step would be to do it in one single loop, instead of having three separate methods that all look the same except for some adjustments to the looping logic. To do this you should have three variables, keeping track of the longest, second longest and third longest string respectively. Then for each string you simply check whether it's longer than the longest, second longest or third longest string and, if so, adjust the variables accordingly.