Panpsychism has a marketing problem before it has a philosophy problem. Say the word and most people picture crystals, or the idea that rocks are quietly brooding. The actual view, defended in serious philosophy journals by people with no interest in crystals, is narrower and more interesting than the caricature: that experience — some minimal, unstructured form of it, nothing like a rock having thoughts — is among the basic properties of matter, the way mass, charge, or spin are basic properties, rather than something that mysteriously switches on once matter gets arranged in a sufficiently complicated way, such as a brain. It is worth taking seriously not because it is obviously true, but because it is a genuine answer to a genuine problem, and its weaknesses are as instructive as its strengths.
The problem it actually solves
Physicalism, the default view that consciousness must ultimately reduce to physical processes, runs into the hard problem: no amount of functional or structural description of a physical system seems to entail why that system has any experience at all. Panpsychism's move is to stop treating experience as something that needs to be derived from non-experiential physical facts in the first place. If some primitive form of experience is already there in the fundamental constituents of matter — however physics ultimately describes those constituents — then a complex system like a brain doesn't need to manufacture experience out of nothing. It only needs to organize experience that, in some minimal sense, was already present in its parts. This is the appeal in one sentence: panpsychism trades the question "how does experience arise from the non-experiential?" for the question "how does simple experience combine into complex experience?" — which sounds like a lateral move, but the second question, unlike the first, at least has a candidate mechanism to investigate: composition, the same kind of relation physics already uses to explain how simple particles combine into complex objects.
It also lets a panpsychist keep something physicalism and dualism each sacrifice something to get: a single, unified physics. Dualism posits two fundamentally different kinds of stuff and has to explain how they interact. Panpsychism posits one kind of stuff — matter — and says experience is simply one of its basic properties, alongside the ones physics already catalogs. Philosophers like Galen Strawson have argued, under the label "realistic monism," that this is in some sense the more conservative option: if you take physicalism seriously — really seriously, as the claim that everything is made of one kind of stuff — and you also take the reality of experience seriously, panpsychism is closer to what falls out than eliminating experience or adding a second substance.
The combination problem
Panpsychism doesn't remove the hard problem. It moves it downstairs, into the question of how simple experience becomes complex experience.
The trouble panpsychism inherits in return is called the combination problem, and it has a genealogy that predates the current revival by more than a century. William James, in The Principles of Psychology (1890), objected to an early version of the "mind dust" idea on the grounds that a hundred separate feelings simply summing together doesn't obviously produce a hundred-and-first feeling that is aware of all the others as a unity — you'd just have a hundred separate, uncombined experiences, side by side, none of them knowing about the rest. Your experience of reading this sentence is not, introspectively, a chorus of a trillion tiny particle-experiences each independently reporting in; it's a single, unified perspective. Explaining how micro-experience could combine into that kind of unified macro-experience — sometimes narrowed to the "subject-summing problem," how many small subjects of experience could ever add up to one new subject rather than just remaining many — is the open technical debt panpsychism has to pay for what it bought in the previous section.
Contemporary panpsychists have tried several routes past it. One, associated with philosophers like Philip Goff, is to flip the direction of composition entirely: instead of starting with micro-experience in particles and asking how it combines upward (micropsychism), start with the cosmos as a whole as the single fundamental subject of experience, and treat individual minds as some kind of partial, differentiated aspect of that one cosmic subject (cosmopsychism) — trading the combination problem for a "decomposition problem" that some philosophers consider more tractable, though it is its own live and contested debate. Strawson's own response is less a solution than an honest admission: that there may be something about the nature of experiential combination we simply don't yet have the concepts to describe, in the same way pre-Maxwell physics lacked the concepts for electromagnetic fields — a promissory note, not a dodge, but a promissory note all the same.
Weighing it honestly
None of this makes panpsychism the obvious winner among theories of mind, and I don't think it is one. Its strength is real: it takes the hard problem at face value instead of explaining it away, and it offers a coherent, monist alternative to dualism that a surprising number of serious philosophers — not mystics, not New Age writers, working analytic philosophers of mind — now consider live rather than embarrassing. Its weakness is equally real: the combination problem is not a minor technical wrinkle to be mopped up later, it is the same kind of gap the hard problem posed in the first place, now relocated to a different junction in the theory. A skeptic is entitled to ask why trading one unexplained transition (non-experiential matter to experience) for another (simple experience to unified experience) counts as progress rather than relabeling.
My own view, for what it's worth, is that panpsychism earns its seat at the table not because it resolves the hard problem but because it is honest about what any resolution will cost. Every serious position pays somewhere — physicalism pays at the explanatory gap, dualism pays at the interaction problem, panpsychism pays at combination. Weighing those costs against each other, rather than pretending one of them is free, is most of what good philosophy of mind actually does. The Stanford Encyclopedia's entry on panpsychism and its supplement on the subject-summing problem lay out the full architecture of the debate in more technical detail than I've used here.
For the problem panpsychism is trying to answer in the first place, see The Hard Problem of Consciousness. For what happens when a related intuition — that experience might be woven into physics at a fundamental level — gets tested against actual quantum mechanics rather than philosophical argument, see The Quantum Observer and Consciousness.