It's not true that entanglement is what makes quantum computers faster. The Gottesman-Knill theorem showed that many quantum algorithms that rely on entanglement can be efficiently implemented classically as well. As the Wikipedia article I linked for the Gottesman-Knill theorem says, it is not terribly well-understood as to where the quantum speedup comes from.
Having said that, the simple reason as to why one would expect entanglement to be the reason behind quantum speed ups is the following: if you have $n$ classical bits, you need $n$ classical bits to describe a possible state of this system. On the other hand, if you have $n$ qubits, you need $\mathcal{O}(2^n)$ classical bits to describe a possible state of this system. However, if you only allow unentangled states of $n$ qubits then you only need $\mathcal{O}(n)$ classical bits to describe a possible state of the system. Thus, the very crude intuition is that if you manage to do computation using entangled states of $n$ qubits, you're somehow juggling around information that would need $O(2^n)$ classical bits to represent -- and thus, you should get a speed up. Again, notice that this is very crude reasoning, for example, even tho you need $\mathcal{O}(2^n)$ classical bits to describe an $n$ qubit state, you cannot encode $\mathcal{O}(2^n)$ classical bits in $n$ qubits to communicate with someone. The idea is that even if you can't extract more than $n$ bits of classical information when you actually read off an $n$ qubit state, clever computation algorithms might be able to somehow use the larger phase space behind the scenes without violating the rule that when someone finally reads the $n$ qubit state, they don't extract more than $n$ bits of classical information.
