Your vote ends up counting for the winner or a loser. With STV the chance of it finally counting for the winner is increased for each additional preference since each time you are counted for a loser your vote is transfered to someone who hasn't lost yet. Until you run out of people on your list or someone is elected.
Now what's the point of having voted for the winner? Hopefully that means you are less unhappy with the results than you could be, this stops being true if your list of stated preferences is longer than your list of real preferences.
Say I like A more than B more than C but couldn't pick D E or F out of a lineup. I Vote A,B,C and leave the rest blank since if my people lose I really don't care who wins. This means if A,B,C are eliminated I abstain from the rest of the election. If I listed D,E,F I would just be diluting the preferences of the people you do care about their differences with however I end up ranking them.
Now say I really detest G. I would put D,E,F on my ballot to be sure I don't abstain in a contest between G and any of the others. The order doesn't matter to me, but it may impact the election if more than one of them is still in contention after my people are out.
Limiting the number of ranked choices costs me my lesser opinions. The assumption is that my lessor preferences are less strong or not as important to me. In my first example this held, but in my second it did not.
Three may be a practical compromise between voting for and against. Or it may be a simplification on the assumption (supported by your comment and my ignorance) that many people may not have opinions on that many people.