The digital numbers are converted to analog so that they can drive speakers and make sound. If you know anything about digital recording, like what the Nyquist frequency is and what anti-alias filters are, then you've got a head start learning about digital synthesizers.
Digital synth have the same functions as analog synth. The difference is that instead of circuitry, every module in a digital synthesizer is a subroutine inside the main sound synthesis program. The program is a loop that repeats one time for every single output sample. The subroutines have to be executed at least one time per sampling period, and some of them have to execute one time per note per sample period. What this means is that the number of notes that can be played, and the sound complexity is determined by how fast a computer it's hooked up to. Just like with computers, a faster CPU costs more money, and synthesizers with faster processing units are more expensive.
Digital synthesizers use digital signal processing routines to make sounds like musical instruments. Electronic keyboards, like acoustic keyboards, make music through generation of sound waves. The earliest digital synthesizer experiments were made with everyday computers as scientists researched the new field of digital sound production.
The earlier commercial synthesizers used hardwired digital circuits to put in place techniques like additive synthesis and FM synthesis. These models became available in the early 1980s. "Wavetable synthesis" and physical modeling were other techniques that became possible with the higher speed microprocessors and digital signal processing technology. The Synclavier was the name of one of the earliest digital synthesizers. The first commercial all-digital synthesizer was the Yamaha DX7. It became a fixture in 1980s music for its amazing versatility.
Some of today's digital synthesizers exist only as software that synthesizes sound by using PC hardware, a fast CPU, and clever programmers so as to get response similar to dedicated synthesizer systems. To reduce latency, which is the period when the note is played and when it shows up, some sound card manufacturers have made special digital signal processing hardware. Digital synths often have the added advantage of onboard accessibility. The switchable front panel controls take care of their own functions. Dedicated synthesizers are also able to best their software-only counterparts with extra functionality as compared to a mouse-controlled system.