The title is pretty much all I want to ask. Why are qubits entangled? To my knowledge (which isn't that deep) a quantum register can be realized without entangling the qubits.
The only purpose of quantum register is to store the qubit. Even though the qubit may be entangled with other qubits, the quantum register can still store and preserve all information of the qubit. So entanglement is not directly related to quantum registers. Anyway, entanglement is required to achieve many effects that cannot be obtained from classical computer.
One of the reason to have entangled qubit is to use the quantum teleportation. It allows you to move the qubit to other computer by communicating through classical channel.
Another reason is the computation speedup. $n$ qubit system have $2^n$ basis orthonormal state in the Hilbert space. To have universal quantum computation (that is, it can simulate all other quantum computer), all possible states must be accessible by some unitary operators. If you restrict to the state with no entanglement, the computation is not universal since you are blocking many evolving path and the states.
Certainly, we can also encode computational problems of size $n$ bits, say $n=32$, into a single particle with $k$ basis state instead. However, it requires an atom $k=2^n$ energy level! We dont have measurement accuracy to distinguish all of them and it means that this approach is not viable at all. On the other hand, we only need $n=32$ qubit with entanglement to achieve the same computation and we can perform somehow accurate measurement in individual qubit one by one. So, entanglement allows scalable quantum computer.