I'd say, as others have said, Fi is kind of a holistic, fuzzy-logic approach to things. Specifically, it is the introverted version of such. Ti's approach is far more deliberately logical and objective, but it is still the introverted version.
The introverted functions are interesting in that they can kind of be regarded as the entity responsible for answering the question, "But what do you really think?" These are the thought processes close to one's heart/self.
I would draw similar distinctions between Te and Fe. Fe is holistic analysis applied externally, Te is more deliberately objective analysis applied to the same. As extroverted functions, they characterize those things we tend to say openly to others, without fear of revealing our true selves.
The Fx functions are "people oriented" because their holistic approach is very well suited to analyzing and evaluating people, whether as individuals (Fi) or as groups or part of a group (Fe). The holistic approach of Fx is not as well-suited to more objective "thingies", where a detailed understanding of underlying logical rules is often more applicable than "fuzzy logic." The analytical approach of Tx is not well-suited to analyzing people: people aren't logical, they don't make sense, they don't seem to be based off of clear underlying principles that explicitly dictate everything about them.
To me more clear about the holistic vs logical, I should note that "holistic" works very well when there is no good way to get a grasp of underlying deterministic principles, that instead one must work with supposition, what things "look like" and characterizing situations/problems in ways that are qualitative but not very quantitative at all. Contrariwise, "logic" works well with good, objective definitions that are always true, and can be quantified.
So it isn't so much that F is "feeling" or that T is "thinking", but that F has a fuzzy, qualitative approach to judging that, in practice, works extremely well with people and makes both Fe and Fi "people oriented." The fuzzy approach can work with other situations, too, but isn't as reliable with respect to objective metrics/standards, just as much as T can work fine with some "people problems" but often make a botch of things.
Fi is specifically using that holistic judgment in an introverted way. At it's best, it is good at understanding oneself as a person, and at understanding other people, in terms of how they differ from oneself. This is precisely where the seeming contradiction of being relaxed and forgiving and nonjudgmental but also quite firm and judging and obstinate with respect to particular beliefs comes from. In order to understand others as individuals, one must accept them as is, that there is an inviolable core of other individuals that must be respected.
Yet in understanding oneself, one's own core is just as inviolable and cannot be violated lightly. One accepts oneself and one's own strengths and flaws, understanding their interdependency (that one cannot be perfect in every way, because different kinds of perfection/strength are mutually exclusive). To that end, a violation of one's "core values" is a violation of one's core, of the kind of person one believes oneself to be. In order to change those values, one must change oneself. It's doable, and Fi users do it occasionally (more frequently when younger and still growing), but one doesn't easily change oneself just because one is told that one is wrong. One must evaluate, understand and accept that the change is right and good, first, which is not a fast process, and then one must work to change oneself, which is also not a fast process.
I don't believe "abstract" is a good way to describe how Fi handles people (though I understand and agree with the thought/idea behind using the word), because it is too easily confused with Ne (or Ni for that matter). I would describe it as more "internalized", or "not directly dependent upon empirical observation." It tends to understand people as individuals, in terms of others' own self-understanding, as opposed to understanding people in terms of how they interact with other people.
My T/F distinction might tend to indicate to some that Fi "can't do math," when nothing could be further from the truth. Math is a learned skill, and I do not believe a young IxFP has to "develop" or "differentiate" inferior Te (or Ti for that matter!) in order to do math. Fi is about how one understands and evaluates reality in order to make decisions. I would imagine that the Fi personal "shortcuts" for doing math differ significantly from those of Ti or Te, though I suspect it would be difficult to measure such a thing. It is, rather, the acceptance that the fuzzy logic judgment is "OK" or even preferable in most cases, that means one "uses Fi."