I personally came to panpsychism from the consideration of a couple of things:
1. The evolution of life on earth from chains of amino acids that somehow formed into RNA with the help of electricity and eventually monocellular organisms, the way these monocellular organisms eventually evolved into complex life and how the more complex life becomes the more "conscious" it also becomes, humanity being not the only animal to experience self awareness, with many animals (not just mammals) demonstrating many of the building blocks of what we consider to be hallmarks of consciousness in human brains. If we evolved from single celled organisms, there is no one point in evolution where we can point to as being a "light switch" between "lacking consciousness" and "having consciousness". There is also no one point in evolution where we can definitively draw a line between "non-life" and "life" because there are things like viruses and plasma clouds in deep space (that self-replicate in double helix shapes) but don't need to respirate or eat.
2. The fact that complex anaerobic life was able to evolve on earth. I watched a David Attenborough documentary about a type of deep sea fish that evolved near volcanic vents that doesn't use oxygen at all, but uses nitrogen. This suggests a form of convergent evolution, but also shows how if complex life doesn't need oxygen, life almost certainly exists elsewhere in the universe from earth, and the relative complexity and decision making capability of this fish shows that "consciousness" can exist in anaerobic organisms.
If life is something that spontaneously and commonly arises all over the universe, that suggests that "life" and "consciousness" are inextricably tied together in a continuum, and in fact at its most fundamental level, "life" and "consciousness" may be synonymous with "awareness of own existence".
3. Complex consciousness such as what humans enjoy is sustained by the interactions of billions of single cells working together, none of them understanding individually what "human consciousness" is and yet contribute to it. In fact each of us when we are conceived started out as a single cell, without "consciousness" as we typically define, and at what point during gestation did we miraculously become conscious? We can't point to a tipping point where one cell more suddenly turned a brain from "lacking" to "having consciousness". It's a continuum. So what is consciousness in humans? If you think it's some special property of human DNA, think about how actually the majority of our serotonin is produced in the gut with the help of billions of bacteria that live within us. Likewise we have more bacterial cells living on our skin, contributing to overall health, than we actually have human cells.
4. The fact that our human consciousness and what we see as a singular ego, is actually the product of billions of cells (not all human) working together without an understanding of the larger whole, suggests that human consciousness is actually a collective consciousness.
This is not unprecedented, the idea of individual pieces coming together to create a larger whole. Even in single cells, what we call mitochondria today or chloroplasts today were themselves once individual cells far back in evolution, but now they are but working parts of modern plant or animal cells.
We evolved from a single cell, and now we are collections of billions of cells, working in concert, somehow with a belief that we are a single ego.
5. If this is true, which it is, then it is the height of arrogance to assume that human consciousness is where the spectrum of consciousness ends.
6. Fractals show repeating patterns to infinite levels of depth. Fractals are not just mathematical concepts, they are real phenomena in the physical world. From the spiral of a fern frond, a seashell, a cyclone, a spiral galaxy, fractals repeat themselves at all levels of the observable micro- macro- and cosmic reality. This gives us a clue for how we can make connections between the world we experience at our level of reality (as humans) and larger levels of reality beyond human comprehension.
An example of one of these patterns repeating is how dark matter is distributed through the universe compared with the human brain.
7. We as humans experience time a certain way, at a certain subjective level of speed. A fly, a dog, a child, a whale, even a plant, all have different experiences of time relative to their age and their lifespan. As such, we are disinclined to see the movement of living organisms whose experience of time are slower than ours, as being truly purposeful. If you show a sped up video of a vine growing, a mould creeping along a surface, they are more easily seen to have a level of "intelligence" even if we wouldn't call it "thought" analogous to human thought.
It therefore follows that if there were higher forms of consciousness, collective consciousnesses such as ecosystems, or even galaxies and the universe itself, the speed of information transfer would occur from our perspective so slowly that we would barely recognise it, or not recognise it at all.
Given the limited human lifespan, and the incredibly short time human civilisation has had technology to measure the universe in comparison to the actual AGE of the universe, it's likely that the information-transfer mechanisms of a "cosmic mind" would occur far too slowly in comparison to human time, for us to properly measure with current scientific methods and technology, and thus it is currently impossible to prove or disprove panpsychism, although it may become possible in the future.
However I do believe that panpsychism can be inferred as a very convincing hypothesis, based on the reasons that I have listed above (as well as others I don't have time to get into now).
Thanks for reading and I hope you can see that panpsychism as I see it, doesn't have to be incompatible with science.