How SAM works
SAM was written to be usable on many different computers. So instead of using the SID chip in the usual way, the CPU has to work to sample the phonemes itself. The SID could have taken a lot of that on, since it has its own oscillators, waveform generators, ASDR volume control, and all that. This would all have been very useful in speech synthesis, but it is not used because the program was written to be portable.
For most phonemes, the program does a tight loop, which gets two sine waves and one rectangular function, each at a different frequency and each scaled by an amplitude, adds these three together, and stores the result (divided by a constant) in the SID's master volume register. That's good for sounding out continuants1. For other phonemes, say the sibilants2, random data and other tables were used instead. And for plosives3, I'd wager the same thing was done, but the amplitude was simply altered with each loop iteration. This arrangement, with the looping, scaling, adding, as you can imagine, takes more than a few cycles on the lowly 6502. And it is very sensitive to timing because even a few cycles here and there will change the resulting soundwave quite drastically.
Fine, but what's it got to do with blanking the screen?
On the C64, if the screen is being displayed4, then every so often the CPU gets halted for some time, to allow the video chip to read the memory faster. Between 40 and 62 cycles (depending on various details) are taken from the CPU in this way every 8 scanlines (again, depending on various details). These scanlines are called badlines, and under normal operation there are either 24 or 25 of them per frame. That means the CPU loses 960-1550 cycles every frame.
The CPU on a European C64 ran at 985 KHz, and so 1000 machine cycles is 1.015 milliseconds. That's about the amount of time, per frame, during which the CPU may not run if it is stopped by the graphics chip, which means those loops I talked about do not run during that time.
When it comes to normal human speech, a phoneme lasts only a very short amount of time. A phoneme can easily last less than, say, fifty milliseconds. Losing even 40 machine cycles puts a considerable dent in that timeslice, so they made sure that wouldn't happen, by making sure no badlines would happen, by making sure no text or graphics would get displayed.
1: continuants are vowels and consonants which can be elongated, like
2: sibilants (a subclass of continuants) are consonants like
th etc. which work by constricting airflow so that it kind of hisses in the mouth.
3: stops (aka plosives), are consonants that involve blocking and then releasing airflow, like
g. These sound very different from start to the finish, and cannot be pronounced for a long time
4: The VIC-II has a bit in register $d011, which blanks the display. It disables all badlines too.