This is basically a marketing term, but one that described a massive shift in technology.
All computers and electronics operate on the theory of boolean logic – 1 or 0, on or off, true or false. By combining a massive number of boolean logic “gates,” you can do an unlimited number of calculations.
(See the book Code: The Hidden Language of Computer Hardware and Software if you want a detailed explanation of how this works. Suffice to say you can do any calculation by combining enough on/off states. See also: Turing Complete)
Throughout history, we’ve needed a way to represent this boolean logic – how do we manage potentially millions of on/off states?
This was first done using the vacuum tube. These were literally glass tubes with all the air sucked out of them (hence the “vacuum” part) and replaced with an inert gas. By heating one end of it, electrons pass through the vacuum, thus completing the circuit.
In 1956, the first computer had 18,000 vacuum tubes. This worked, but it had problems. The tubes use the physical mechanism of heating, which means they can burn out – if air seeps in, the heating element burns out (this is the same reason light bulbs are vacuums). Clearly they also used a lot of electricity, and presumably generated a lot of heat.
The vacuum tube was replaced by the transistor, invented at Bell Labs. Transistors use silicon to control how electrons move. There are no moving parts, and no physical action (though, to be fair, there were no moving parts in a vacuum tube, either, but it replied on the physical concepts of vacuum and the interaction of the parts)
All the logical working of a transistor are achieved in a solid state, not though a gas. This means they’re smaller and use less power. The average computer chip today has billions of transistors.
When transistors became popular, the phrase “Solid State” came into vogue as a marketing term to identify devices which used transistors instead of vacuum tubes. It has a catchy alliteration to it, and in the 1960s, when devices like radios and alarms clocks began to use transistors, it was a trendy way to signal a higher level of technology. Kind of like how “fuel-injected” was once a trendy badge to put on cars.
Today, it means nothing because everything uses transistors. All electronics and have effectively become solid state.