Like Amethyst said, the main purpose of these moves is to stop your special attacker from getting walled by Chansey and Blissey. (And other special walls. Pretty much all Special Walls have less Defense than Special Defense, so they will take more damage from Psyshock.) These two Pokémon specifically are ridiculously bulky on the special side, I mean, look at this:
252 SpA Mewtwo-Mega-Y Psychic vs. 4 HP / 252+ SpD Blissey: 117-138 (17.9 - 21.1%) -- possible 5HKO
And this is why moves like Psyshock are good:
252 SpA Mewtwo-Mega-Y Psyshock vs. 4 HP / 252+ Def Blissey: 352-415 (53.9 - 63.6%) -- guaranteed 2HKO
Even though Psyshock has worse Base Power than Psychic (80 vs 90), it does significantly more damage. Also, being able to hit Defense and Special Defense with Special moves gives you the benefits of a Mixed attacker, (A Pokémon that uses both Special and Physical moves) with none of the drawbacks.
Also, the base power difference isn't that high. Psyshock and Psychic are only 10 BP away, and Psystrike actually has 10 more BP than Psychic. Secret Sword is 5 BP more than Aura Sphere, the only other good Special Fighting-Type move. (Although when Secret Sword was introduced, Aura Sphere had 90 BP instead of 80.)
One more thing that Psyshock, and moves like it do, is break bast Pokémon using Special Defense boosting moves, like Calm Mind. At least in my experience, Calm Mind is more common than Bulk Up, or Coil, because much more Pokémon learn Calm Mind, and often the Pokémon that do learn Bulk Up, etc also learn Swords Dance, which is usually better. Because Calm Mind is so common, moves like Psyshock can really help against Calm Mind users like Raging Bolt and Magearna.
I think that about sums it up. If I forgot anything, comment so I can fix it. Hope I helped.
