That filter is basically a low pass filter that filter out whats above a certain frequency. It dose work without it but it will improve the audio quality. That filter on the DAC is not as inportant as the one on the ADC but it dose inprove the quality a little.(It will help a lot with very low sample rates)
What this filter dose is filter out the frequencys too high for the sample rate to capture. If you leave them in they become noise. As for the output it smooths out the signal, because the signal is slightly chopy because of the sample rate and bit depth.
With all this you should be able to produce crystal clear sound with only a dsPIC (And some kind of amp chip if you want to drive headphones or speakers directly from it)
Also one important thing is a clean power supply. Noise on the power supply will cause noise in your analog input and output.
Also note that an SD card is NOT a 5V part. It works on 3.3V , this is also why i recommended a dsPIC33F since they work on 3.3V also and you can get them in PDIP so you can put the stuff together on a breadboard.
Oh and i hope you know how to program in C, since to use the FAT filesystem lib you will need to compile it with MPLAB C30 compiler, so that means you will also have to write your code in C.