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i want to know the different among 아서 and (으)니까 and 기 때문에

I think that It is not easy but good question. Following is my answer.

~아서:
1. 동사A+아서+B: B is done by doing A.
ex) 가서 본다 : I see by going there.

It is used to represent sequential motions often. For example,
ex) 잡아서 먹는다. I eat it by catching it.
-> I catch it, and eat it. (sequential motions: catch and eat )

2. 형용사+아서: close to "~, so~".
It is used to represent reason and result.
ex) 작아서 먹는다. It is small, so I eat it. (reason and result: because it is small, so I eat it)


~(으)니까:
1. 동사+(으)니까: close to "~, so ~" or "~, and ~".
It is used to represent an action and a coresponding consequence. Usually, the action is the reason of the consequence apparently, but sometimes the speaker is not sure about the actual reason. For example,
ex) 잡으니까 도망갔다. I caught it, and it ran away.
(but the speaker is not sure whether it ran away because I caught it)
(action: caught, consequence: ran )

2. 형용사+(으)니까: close to "~, so~".
It is used to represent reason and action.
ex) 작으니까 먹는다. It is small, so I eat it. (reason and result: because it is small, so I eat it)


~기 때문에 : it means "because" always
ex 형용사) 잡기 때문에 도망간다. it runs away because I catch it.
ex 동사) 작기 때문에 먹는다. I eat it because It is small

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comparison)
(동사)+(아서/(으)니까/기 때문에)
잡아서 먹는다. :먹는다 (how? -> 잡아서) I eat by catching
잡으니까 도망간다. :도망간다 (what after? ->잡으니까) I catch it, then It ran away.
잡기 때문에 도망간다. :도망간다 (why? -> 잡기 때문에) It ran away because I catch it

(동사)+(아서/(으)니까/기 때문에) : always "~ because"
작아서 먹는다 : I eat it because it is small
작으니까 먹는다 : I eat it because it is small
작기때문에 먹는다. : I eat it because it is small