Learn Hardcore Portuguese (Brazil): Eu fiquei em casa ontem à noite porque estava cansado. - I stayed at home last night because I was tired.

Eu fiquei em casa = I stayed at home (I remained at home).
Eu estive em casa usually means I was at home (at some point), often more like “I happened to be there / I was there,” not emphasizing “staying in.”
Eu estava em casa describes a state/background (“I was at home”), but it doesn’t highlight the decision/action of staying the way fiquei does.

What tense is fiquei, and what does it imply about the time?

Fiquei is the pretérito perfeito (simple past). It presents the action as completed and tied to a specific past time: ontem à noite (last night). It suggests that you did stay home during that time period.

Why does the second verb change to estava instead of estive or fui?

Estava is pretérito imperfeito (imperfect) and is commonly used to give a reason/background condition in the past:

...porque estava cansado = because I was tired (as a continuing state at that time).

Using estive cansado is unusual in Portuguese; tiredness is typically described with estar in the imperfect in this kind of explanation. Fui cansado is not used to mean “was tired.”