I think your question itself (especially the edit) reveals the answer: you want to know what it (everything) looks like from the inside of the box, you want the perspective of someone inside the box. If I understand you correctly, you’re not interested in the box itself, or even (only) what is inside the box, but in everything that can be seen from inside the box, which may include things both inside and outside of the box. The preposition “from” which you quite naturally used in formulating the question seems essential to retain in your final sentence: “I want to know what it looks like from inside the box.” Note also that you said twice “I want to know what it looks on the inside,” which is not grammatical English. Either say “what it looks like” or “how it looks”, but not “what it looks”. As to the first sentence in your original question, I agree with BadZen that saying “I want to look inside the box” indicates an interest in either the box or its contents, not the perspective of being inside the box, so that is not the formulation you want to use to express the idea you clarified was your actual intent. Of course, [the box] can be replaced by any place or situation.

  I want to look inside of the box.

^ This means you want to open up the box and gaze into it, and at its contents as well. You might also mean that there is a specific thing you are looking for, and you want to see if that thing is inside the box.

  I want to know what the box looks like on the inside.

^ This means you are curious about the appearance of the box itself. You might ot might not actually want to open it to see, though. You're not interested in looking /for something/, here.

  I want to look on the inside of the box.

^ This is incorrect, there's no need for the preposition phrase starting with "on". Just say "I want to look inside the box."