Why do some verbs take 'sein' in the perfect tense and how do I figure out, whether they do or not?

Verbs that take 'sein' in the past are called intransitive verbs and there are certain questions that you can ask yourself to figure out, whether a verb takes 'sein'. The first question is "Does the verb indicate a movement from one place to another?". If so, it takes 'sein'. Verbs like 'gehen' (to go), 'fahren' (to drive/go via transport) and 'reisen' (to travel) all show a movement from Point A to Point B. The second question is "Does the verb describe a change of state?". If so, it takes 'sein'. Verbs like 'sterben' (to die), 'einschlafen' (to fall asleep) and 'umziehen' (to move house) all show a process of going from one state of being to another; e.g. I am alive -> I die -> I am dead; I am awake -> I fall asleep -> I am asleep; I live in house A -> I move house -> I live in house B.

The last question to ask is "Does the verb directly affect anything else in the sentence?". If so, it does not take 'sein'. This might seem like the first question to ask but this requires a bit more thinking so I always leave it till last. Verbs like 'blieben' (to stay), 'passieren' (to happen) and 'sein' itself all take 'sein' because they do not affect anything else in the sentence directly (i.e. without using a preposition). For example, neither 'I stayed' nor 'that happened' need anything else to make sense and only affect things indirectly via prepositions. The reason 'sein' is included here is because it always refers back to the subject; e.g. in 'I am a dog', both 'I' and 'a dog' are affected by 'am'. It's a loop, not a sequence. Remember, you can always check the dictionary if you're unsure.