6 ideas
A strange loop is a level-crossing feedback pattern in which a system's high-level symbols come to represent the system itself, so that moving through levels of abstraction returns you to where you started. The 'I' is this kind of loop: the brain perceives itself and reifies that perception into a symbol, and that symbol then drives the brain's own behavior.
Gödel showed that Principia Mathematica, a system built only to talk about numbers, could be made to encode statements about itself through a numerical coding of its own formulas. By analogy, neurons that mechanically process signals can, at a higher level of description, carry meaning about the very organism they compose, so self-reference arises without any extra ingredient.
The 'I' is a useful illusion, a coarse-grained summary the brain constructs because tracking trillions of micro-events is impossible, much as we perceive a 'marble' in a box of envelopes that is really just their stacked edges. It is not a separate thing, yet it has real causal force, because high-level patterns are the right level at which to explain behavior.
High-level patterns such as beliefs and desires legitimately cause physical events, even though every step is also fully explained by particle physics. Saying 'the idea made me act' is more explanatory than tracing the particles, just as a domino chain can best be explained by the primality of 641 rather than by the physics of the dominoes.
After his wife Carol died suddenly of a brain tumor, Hofstadter looked at her photograph and felt that part of her perspective still ran inside his own brain. Spouses build such rich internal models of each other that a lower-resolution copy of one person's loop can continue after death, which he used to reframe grief and continued existence.
Consciousness is a gradient that scales with how rich and self-referential a creature's perceptual loop is, not a switch that is either on or off. A mosquito has almost none, a dog more, and a mature human the most, which Hofstadter marks with the playful unit 'hunekers'. That ranking guided his moral choices, such as vegetarianism and weighing lives unequally.