"Él mantiene a su hijo."
Translation:He supports his son.
I will try to make it simple first and then complex.
Regarding the verbs:
mantener is to hold, to sustain (sth) although can be used figuratively, like to provide means of sustainment, i.e. money thus the expression "ser un mantenido" somebody who does not work and gets money from family or friends and lives off them.
Apoyar is to lean on (sth) it can be literal or fig too.
Very often, the spanish expression is more of:"I lean on you" than "you're my support" and when using the name, "eres my apoyo"
From the technical perspective apoyo is a support, but it's taken from the perspective that "is receiving load" and not "supporting" it