How is interpolation performed when a sampled sequence is upsampled?
Conceptually, inserting zeros into a sequence of samples effectively creates an amplitude-modulated train of narrow pulses. In the frequency domain, this looks like many copies (or "images") of the original baseband signal, repeated at intervals corresponding to the original sampling rate.

Passing this sample stream through a low-pass filter eliminates most of those extra copies, leaving only the ones that correspond to the new sample rate. In the time domain, the sample stream is now a "smoother" representation of the original signal.

The ideal filter for this purpose is the so-called "brick wall" filter, which has a flat passband and the fastest possible cutoff. In the frequency domain, this looks like a rectangle. In the time domain, the impulse response is the sinc() function.

There is no new information in these new samples that wasn't in the original set of samples, but now the signal is easier to handle in certain ways, particularly with regard to D/A conversion and analog filtering.