Back in the old days, it was actually illegal
to connect non-telephone company equipment directly to your phone line. The telephone company wanted to ensure that the proper operation of their network was not disrupted by unapproved devices.
Since gaining telephone company approval for devices connecting directly to the network was difficult, and because customers had to obtain permission to hook into the network, the first modems used acoustic couplers to connect to the telephone network. You could obtain direct-connect modems, but they were very expensive and you had to convince Ma Bell to let you plug it in.
An acoustic coupler consists of a speaker and a microphone inside seperate rubber cups designed to hold a standard Western Electric telephone handset. The speaker on the coupler sends signals into the microphone of the handset, and the microphone receives signals from the handset's speaker.
This arrangement is perfectly adequate for sending signals at the data rates common at the time: 110 and 300 bps. They could also operate at 1200 bps, but once you start getting into 2400 bps and above, acoustic couplers become unreliable due to limited bandwidth and variations in the quality of telephone handsets.
With the breakup of the AT&T telephone monopoly in 1984, individuals could now purchase their own telephone equipment that connects directly to the phone line, including direct-connect modems. Not only were direct-connect modems free from the noise and bandwidth limits of acoustic couplers, but now the modem could handle call setup and teardown on its own, which was a lot more convenient to the user. This made the creation of Bulletin Board Systems (BBSs) a lot easier.
Acoustic couplers are a curious artifact of history that resulted from corporate politics and the unwavering monopoly of the Bell System. However, they are still the only way to do data from a payphone!