Fi is pretty subtle.
It's one of the "perceptual" decision making processes (along with Ti), which means it's good at making decisions in the middle of massive, nonlinear hits of unstructured information (Te and Fe need their info structured to deal with it).
But unlike Ti, it's more personalized; it encourages a personal relationship with the evolving information. But a prerequisite for any personal relationship is prior experience to determine what's acceptable or not.
An *extremely* simple-minded example of how Fi works is cooking. Say you're making sauce. You need to taste your sauce as you're cooking it and add whatever ingredients you need to make it taste "good". There's no way to structure that kind of information spontaneously, and Fi and Ti don't need to.
Even though you might say that you're taking in the information via S, you're making decisions on the fly in terms of what will make the sauce "good" or "bad". You need to personally know the difference between good and bad beforehand via prior experience. And you need to know it in your bones. Ti lacks this personal relationship in deciding what to do with sense-impressions, so, to Ti, the terms "good" and "bad" are meaningless in this context.
But Fi naturally discriminates between "good" and "bad" like that -- in terms of personal experience. It strikes other types as odd because, since it's perceptual, it's nonverbal, receptive, and guided by perceptions it can't quite explain. And on a much larger scale than cooking sauces, it derives its principles from prior experience of being human.
An added benefit of being perceptual (Fi/Ti) as that it doesn't need to take things apart so finely; Fi has the capacity to see things as a whole, apart from the assumptions we've been socialized into accepting. It can then reason wholistically to determine the integrity of our actions. Fe has a stronger emphasis on the prevailing social norms, and thus has a harder time providing this kind of wholistic decision-making.
But a negative is that it's very hard to express the nonlinear/nonverbal machinations of Fi's decision-making process. Combined with the emphasis on what's personally good or bad in terms of fundamental human values, this difficulty to clearly and linearly lay out one's line of reasoning to others can be interpreted as being dogmatic and polar, and their deeply-held human values aren't always self-evident to other people.
On the other hand, the emphasis on personalizing perceptions gives Fi a remarkably strong point of reference. Take a tragic play, for example. The emotional impact and meaning of the play are entirely dependent on whether or not we have a strong inner reference shaped by the experience of being human -- the kind that bypasses all the social constructs and strikes on what is essentially human. This gives people who use Fi an amazing amount of empathy.
I think others are bothered by Fi because it can appear so dogmatically certain about things that strike others as relative -- concepts like "good" and "bad" -- yet it has difficulty expressing its justification.
That's how I understand it anyway.