My grandparents made this recording in the late 1950s. If you listen, you can easily hear the problem, but I don’t have the words to explain what’s wrong, which makes it harder to fix.
Any advice would be greatly appreciated!!
That’s some very-bad distortion and I don’t think it can be fixed. 
Do you have the original analog? (Tape, I assume?)
If you can understand what’s being said (often the brain is the best “filter”) it might be best to type-up a transcript to go-along with the recording, and then select the bad sections and use the Amplify effect or the Envelope Tool to reduce the volume of those parts so it’s not so annoying.
That looks like distortion caused by ADC (analog to digital converter) overload—when the analog signal is too large, the digital signal “wraps around” to the other side. That is, when the analog waveform goes too positive, the digital value goes negative, and vice versa:
If you still have the original analog medium (as DVDdoug mentioned), try converting it to digital again but at a lower analog volume.
Agreed 
I was bored so I attempted a fix. It’s not perfect (some overloads are missed and some non-overloads are incorrectly detected as overloads) but it’s better than the original overloaded version:
Here’s the quick and dirty C program I wrote to “fix” it: Fix test-fix.mp3 · GitHub (it takes raw 16-bit signed stereo data, in host byte order, on standard input, and outputs raw data in the same format).
The real fix is to re-convert from the original analog source without overloading the ADC (if that’s what the problem really is).
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