Learn Hardcore Korean: uija saie yeonpiri tteoreojyeo isseoyo.

사이 is a noun meaning “space” or “interval.”
Adding the locative particle -에 → 사이에 turns “space” into “at/in that space.”
So 의자 사이에 = “between the chairs” (i.e. in the space between them).

Why is there an -이 after 연필 (making 연필이)?

이/가 are the Korean subject markers. You attach:

-이 if the noun ends in a consonant (like 연필)
-가 if the noun ends in a vowel.
Since 연필 ends with ㄹ, we use 연필이 to mark it as the subject of the verb 떨어져 있어요.

What is 떨어져 있어요, and how is it different from 떨어졌어요?

떨어져 있어요 comes from combining:

떨어지다 (to fall)
the stative-resultative construction -아/어 있다 (to be in a state)
So 떨어져 있다 = “to be in a fallen state,” and in polite present tense you say 떨어져 있어요 = “(it) is lying there after having fallen.”
By contrast, 떨어졌어요 is simply the past tense of 떨어지다, meaning “(it) fell.” That form reports the action, not the current state.

Could I just say 의자 사이에 연필이 있어요 without 떨어져, and would it mean the same thing?

You can say 의자 사이에 연필이 있어요 to mean “There’s a pencil between the chairs.”

This tells somebody where the pencil is.
But 떨어져 있어요 adds the nuance that the pencil has fallen there and remains on the floor.

Is 사이에 interchangeable with 사이에서 here?

No.

사이에 = “at/in the space between,” used for static location.
사이에서 often means “among” or “while/amongst (an action happening in the midst of).”
Since you want to describe where the pencil is, you use 사이에.

Why can’t I simply say 떨어져요 instead of 떨어져 있어요?

떨어져요 is the plain present of 떨어지다, meaning “(it) falls” or “(it) will fall.”
떨어져 있어요 (with -어 있다) describes the resulting state: “it’s fallen and remains there.”
To talk about the pencil’s current condition (lying on the floor), you need 떨어져 있어요.

How do you form 떨어져 있어요 step by step?