How do spawning algorithms work in games like phantom forces and how could a person make one?

You start by disabling character autoloads. https://developer.roblox.com/en-us/api-reference/property/Players/CharacterAutoLoads

Next you can use the player:LoadCharacter() method to load the characters you want or need (although phantom forces uses a pseudo character iirc. https://developer.roblox.com/en-us/api-reference/function/Player/LoadCharacter

Finally you can determine random spawn locations using math.random and vector3.new https://developer.roblox.com/en-us/api-reference/lua-docs/math

https://developer.roblox.com/en-us/api-reference/datatype/Vector3

If you want truly dynamic and random spawning you can also use some trickery with multiple spawn zones determined by multiple uses of math.random. Further if you only use math.random for the x and z component you can potentially use raycast from some high altitude to find the ground position (for assuming an uneven surface) so you don't spawn people in the sky.