I will try to give a high level interpretation of what is going on during Quantum Teleportation.
We first begin with Alice and Bob each holding 1 bit from a Bell State, otherwise known as a maximally entangled state. At a high level we can think of the entanglement "connecting" the two bits of information and correspondingly the information possessed by Alice and Bob.
If Alice wishes to communicate another bit of information to Bob what is a strategy she may use? Well she could just walk over with the bit, or send it as a quantum signal on a conducting line, but these have the difficulty of requiring quantum communication; using entanglement we can devise a better strategy to communicate the state from Alice to Bob.
Again at a high level, Alice can first entangle her bit to the bit she has of the shared Bell state between herself and Bob; which now creates 3 entangled quantum bits. In many ways the information is already "connected" to Bob, however we still need a way to make that information readable to Bob. If after the entanglement Alice measures her 2 bits; then the bit in Bob's possession will collapse to a single qubit (can be calculated through partial traces easily).
Here we get into the specifics of the protocol, but as your question suggests you have already gone through the "calculations". By sending her measurements as classical bits to Bob, Alice enables Bob to perform rotations on his state and obtain after the rotations the original state held by Alice. I won't explain why/how this works, you can investigate further for yourself.
At its core the entanglement is "connecting" two geographically separated users, and allowing them through further entanglement and collapse to communicate information through their entanglement. This is a fundamental property of quantum information, and of quantum channels.