Why A solid-state relay does the job of an electromechanical relay, switching a load on and off under control of a separate input signal, but with no moving parts. Inside, the control input typically drives an LED that optically couples to a phototransistor or photo-triac, which in turn gates a power transistor, MOSFET or TRIAC to switch the load. The optical coupling gives the same input-to-output isolation that a relay coil and contacts provide mechanically, which is why this topic sits with optoelectronics.
Watch out The choice about transistors driving a relay coil describes an ordinary relay driver circuit; there is still a real coil and real contacts, so it is not solid state. The latching mechanical relay is just a different style of electromechanical relay.
Solid state = no moving parts. Same relay function, semiconductors and an optocoupler instead of coil and contacts.