So I understand that it's "me quedo" instead of "yo quedo" because the "yo" is omitted. Just like everything else in Spanish, the verb tense gives away what the pronoun is.
Why is it then "yo me quedo" instead of just "yo quedo"?
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"Quedar" is an intransitive verb, which can also be use as pronominal (quedar(se)).
This pronominal use means roughly that their subjects are acting upon themselves (I do this action upon myself).
Hoy [nosotros] nos quedamos en casa.
"We" receive the action of staying at home.
Pronominal verbs must be conjugated with a reflexive pronoun (Yo me quedo; Nosotros nos quedamos).
Pronominal verbs fall into three major classes based on their meaning:
reflexive: Yo me lavo (a mi mismo)
idiomatic (reflexive actions per se): Yo me divierto.
reciprocal: Ellos se besan. Las niñas se peinan (one to another. This one could actually be reflexive too. Each girl does her own hair, as opposed to each girl doing each other's hair).
Yo me lavo (a mi mismo ==> reflexive)
Yo lavo los platos
Quedar(se) is a little bit tricky because it is an intrasitive verb, which can be used as pronominal. It does not literally say "I place myself at home", but the agent of the action is simultaneously the patient, which is what a reflexive verb implies.
means "be," while
El museo queda lejos del hotel. The museum is a long way from the hotel.
Nos quedamos en el hotel tres días. We stayed at the hotel for three days.
Los niños no pudieron quedarse quietos justo antes del recreo. The children could not stay still just before recess.
Me quedé en casa para cuidar al gato. I remained at home to take care of the cat.