The purpose of filtering in A/D and D/A conversion isn't at all intuitive.... so perhaps a proper explanation would be in order.
I'm sure you're familiar with the basic fact that a digital audio file can ONLY be used to store information up to just under the Nyquist frequency - which is 1/2 of the sample rate. For a 44.1 kHz file, the Nyquist frequency is 22.5 kHz, which is why a CD can't contain any information above that frequency (and using 20 kHz as the cutoff frequency does give a tiny safety margin there).
However, in reality, any source is going to contain some information above 22 kHz; which will include high order harmonics produced by some instruments like cymbals, as well as actual noise from equipment like preamps and other electronics, hiss from analog master tapes, and even noise present in the room where the recording was made. The problem is that this extra "unrecordable" content doesn't simply disappear when you feed the content into your ADC. In fact, the opposite is true; if there is any content present above the maximum frequency that can be encoded "properly", it is converted into a very audible and unpleasant distortion during the encoding process. The actual process involved, and the exact results, are somewhat complicated, but the net result is that a significant portion of it is "folded back around the Nyquist frequency" into the audio INSIDE THE AUDIO RANGE OF THE ENCODED AUDIO.
Let's take our CD as an example: the sample rate is 44 kHz, and the Nyquist frequency is 22 kHz. Now let's assume that, mixed in with my "audible source material", there is a 28 kHz tone (it could simply be some ultrasonic harmonic of some instrument, or just some 26 kHz hiss in the microphone preamp). If I were to feed that source into an ADC which lacked the proper filtering, that 28 kHz ultrasonic would be "folded down around the Nyquist frequency". This means that most of the energy contained in that 28 kHz tone, which starts out being 6 kHz ABOVE the Nyquist frequency (22+6 kHz), and which cannot be encoded into our output file, will be converted by the encoding process into an equivalent amount of energy 6 kHz BELOW the Nyquist frequency (22-6 kHz). So the energy from our ultrasonic 28 kHz noise source, which was totally inaudible, will appear in our encoded file as noise at 16 kHz. This process will occur with all signal energy which is present in the source that is above the Nyquist frequency - and will essentially end up as noise/distortion in the final file. The actual process is somewhat "messier" than my simplified explanation, so that 28 kHz tone will actually cause noise spikes at other points inside the audible range. And, of course, I used a single tone as an example, while the reality is somewhat more complicated. (Think of it sort of like a distorted and inverted reflection in a camera lens interfering with the desired image.) The purpose of the filter in the ADC is to remove any content outside the frequency range which the encoder can handle properly so this doesn't happen (which means anything above the Nyquist frequency).
Unfortunately, most real-world recordings contain all sorts of energy above 20 kHz, ranging from harmonics of actual instruments, to room noise and such, and even to analog distortion products. ANY of this which the filter fails to remove will end up as distortion and noise inside the audio band in the resulting encoded file, which is why it is critical that the filter remove it - or at least reduce it to a very low level.
(Oversampling avoids this issue by shifting the Nyquist frequency upwards, thus making it easier to design a filter that is flat to all audible frequencies, yet still has sufficient attenuation above the Nyquist frequency. In order to make a "clean" conversion at 44 kHz, to the level required for a CD, you would need a filter that is flat to 20 kHz, yet down about 80 dB at 24 kHz, which is very difficult to achieve in practice. To get a similarly noise and distortion free output at a 96k sample rate, the filter would have to be flat to 20 kHz, and down 80 dB at 50 kHz, which is a far gentler slope, and so much easier to design and build.)