Pleb is just short for plebeian (a commoner, in ancient Rome). In other words, a regular person.
The chain thing comes from block-chain.
Hope that helps.
Discussion
What you have here is a culture that embraces the idea of treating everybody equally without giving any special privileges.
We’re all the same.
It’s basically an anti-diva movement, that values hard work and honesty.
That’s what the “pleb” and “stay humble” slogans try to capture.