This site talks about how the reconstructions were made. Here's what it says about why there are missing episodes at all:
...By the 1970's the show had moved to colour and the old 60's black and white episodes were becoming less popular to show as more and more foreign TV stations opted to show the newer colour stories instead. The copies distributed abroad were supposed to be either destroyed or returned to the BBC after use. Believing that the BBC film archive held copies of all Doctor Who stories, BBC Enterprises decided that their old black and white film prints would no longer be required for overseas sales and their destruction began in the early 1970s.
It was later discovered that the film Archive did not keep copies, all the original prints and negatives had been passed to Enterprises. By the late 1970s virtually all the 60s episodes had been destroyed. Copies of many episodes have subsequently been found in other BBC archives and returned from foreign TV stations and private collectors. ...
...and here's what it says about why we have audio from all of them:
Although the BBC kept some of the sound effects and music used from the 60s episodes, the full soundtracks were not recorded separately. However, several fans did record the stories on reel-to-reel audio tape from their televisions upon their original transmission, most notably David Holman, Richard Landen, David Butler and Graham Strong. Most of the audio recordings that were taken had the opening and closing credits cut out at the time of recording to save on tape, so although all the episodes do exist on audio, there are a few seconds at the start or end of some episodes that are missing. For this reason it is often necessary to combine several different versions to produce a complete soundtrack and the exact start/finish points of episodes are often difficult to determine.
Presumably this also helps explain the quality issues.