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 a 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. Anyways, entanglement is required to achieve many effects that cannot be obtained from a classical computer.
One of the reasons to have an entangled qubit is to use quantum teleportation. It allows you to move the qubit to the other computer by communicating through the classical channel.
Another reason is the computational speedup. An $n$ qubit system has $2^n$ basis orthonormal state in the Hilbert space. To have universal quantum computation (that is, it can simulate all other quantum computers), all possible states must be accessible by some unitary operator. If you restrict to the state with no entanglement, the computation is not universal since you are blocking many evolving paths 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 $n=32$ qubits with entanglement to achieve the same computation and we can perform somehow accurate measurement on individual qubits one by one. So, entanglement allows the construction of scalable quantum computers.