The short version
300 baud is the maximum rate that the telephone line could support using modulation and demodulation technology that was easily available at the time. The specific number is, ultimately, related to the ordinary frequency range of sounds that can be heard by the human ear. The phone line was designed and optimized so people could speak normally and be understood on the other end.
How does this work?
Telephone lines carry electrically encoded audio. That is to say, there's a circuit that turns sound into electrical impulses, which are sent over the phone line. What frequencies need to be carried is determined by how humans speak. Whatever language you speak, the maximum frequency is determined by what frequencies the human ear can perceive.
A modem (modulator/demodulator) converts digital data into audio, which is then sent over the same phone line.
Early modems used old-style telephones with rubber cups that had speakers in them. One cup had a speaker and the other a microphone. You'd set your phone in them and it would physically make audio sounds that were converted by the phone into the electrical signaling required for the phone line.
Later on, the extra step was eliminated and the modem would plug directly into the phone line. It had the circuitry in it to emulate the same procedure, including the part that the telephone receiver did.
If you think about sending digital data over an audio channel, you may think something like Morse code. ITU modems (the ones that most people used that ran at 300 baud) didn't use Morse code, but there was a different encoding scheme called frequency shift keying that worked the same. FSK is much more difficult (generally impossible) for humans to encode and decode by hand, unlike Morse code, but faster. For reference, 100 words per minute (WPM) of Morse code is, "pretty good," and correlates to 75 baud.
Right, but still, why 300?
The speed you can use is based on what you can modulate or demodulate (often one is easier than the other), what your line can handle and how clean (error-free) your line is. The bandwidth on original analog lines was fairly limited. The rubber cups made a pretty good connection to your phone but it wasn't super fantastic. Further, the world of digital to analog conversion was relatively new (compared to now) so less was known about how to do it efficiently and cost-effectively.
Like any system, the maximum rate is determined by the maximum rate achievable by the slowest part of the system.
After weighing all the factors, the ITU published a specification for inter-device communication. For a long period of time, the quality of the existing phone lines and other factors put tolerances in the range of 300 baud FSK range.
Adaptive rate protocols were newer at the time these first standards were rolled out. Fixed-rate systems or user-configured rates (you agreed with your friend that you were both using 300 baud) were more common, although by the end of the phone modem, adaptive rates became the standards. If you knew your line and your modem could handle at least 300 baud, you used 300 baud.
The frequency of speech (people talking to each other) is determined by the frequency response of the human ear, figured out over thousands of years of human society. The bandwidth of the phone line was determined by the frequency of speech. The bandwidth of the modem was determined by what could be sent over the phone line. The symbol rate was determined by the available bandwidth.
It is my personal belief that the actual math (FSK symbol rate calculations, etc.) is fairly pointless unless you're trying to re-design this system, which has already been designed pretty well. Given the audio response of the ear and the mechanics of FSK, 300 was the closest round number.