The following is a simplification.
An instructive way of conceptualizing what an digital oversampling filter does is to picture the unfiltered D/A output spectrum. The music signal occupies repeating pairs of upper and lower side-bands symmetrically centered on multiples of one of the native sample rate (0, 1, 2, 3, etc.). These sample rate multiples begins at zero. So, the first signal band begins with it's upper side-band, and runs from D.C. to 22kHz. This is the signal band we actually utilize for listening. There is an mathematically implied lower sideband, but it would run entirely beneath D.C., so, it doesn't physically exist.
The next pair of signal side-bands are centered on 44.1kHz. The lower side-band band runs from 22.05kHz (44.1kHz - 22.05kHz) to 44.1kHz. The upper side-band runs from 44.1kHz to 66.15kHz (44.1kHz + 22.05kHz). This pattern repeats at each multiple of one of the native sample rate, so the next side-band pair is centered on 44.1kHz x 2 = 88.2kHz.
It's important to recognize that even though the D/A output appears discrete when viewed in the time-domain on an oscilloscope (typically, though not necessarily, appearing stair-stepped) it none-the-less contains a fully devloped analog representation of the signal. It's the many repeating copies (images) of the signal which gives the signal it's discrete appearance. The job of any reconstruction filter, be it digital or analog, is to remove all the repeating image bands, after which, only the expected smooth appearing signal will result.
Oversampling reconstruction filters (also known as, multirate filters) greatly spread out (shift up in frequency) the images by centering them on higher multiples of the native rate. For example, an x8 oversampling filter will center the side-bands at multiples of 8 times the native sample rate (0, 8, 16, 24, etc.). So, instead of the first side-band pair images being centered on 44.1kHz they're centered on 44.1kHz x 8 = 352.8kHz. The next pair are centered at 44.1kHz x 16 = 705.6kHz, etc. The further up in frequency are shifted the image bands the easier it is to completely remove them with a relatively gentle analog filter.