Consider how the number of states are calculated: If you have n qubits, at the time of making the measurements those can be in 2^n distinct (basis) states. This is similar to how in the classical case you can use n bits to express at most 2^n different states.

Which one of the basis states we end up measuring depends on the squared of the complex amplitude of that state (Born rule). Therefore, if you were to simulate a quantum computer, in the general case you would need to keep track of these complex amplitudes belonging to the states, every 2^n of them. Because of the real and imaginary parts this amounts to 2*2^n=2^(n+1) numbers.

Because the State Space of an n-qubit System is the Tensor product of the single qubit State Space.