Transformers only work with AC currents; the principle of operation relies on changing magnetic fields generating electric currents - static fields don't generate a current, so AC must be used. They're used to change one voltage to another (used in power supplies), or to change one impedance to another (often used in amplifier output stages).
Normally electronics require DC current, so this AC current must be rectified. There are many ways to perform this, the most common is with a device called a bridge rectifier. It uses four diodes to move the 'bottom' part of the AC wave into the 'top'. You still don't have DC, but the voltage is 0V or greater at all times. Various degrees of filtering would be applied to smooth this out into something close to DC. The simplest approach is to use a large filter capacitor directly across the output of the transformer. Complexity goes up from there, with the next step up being an LC filter, then a pi (CLC) filter, and so on, with a regulator of some type often added afterwards if line regulation is required.
The
article on rectification at Wikipedia has some good diagrams that might help.
Replacing electrolytic caps might be a good idea if you're up for the effort. If the keyboard is working fine though, you probably don't really need to. Try to avoid mucking with parts in the audio section of the circuitry; I don't know what vintage this keyboard is, but many early electronic keyboards have significant analog signal processing that might be affected.
If you have a multimeter, you can measure the output of the transformer (remember to use ACV mode), and select a transformer that is around that range. 37VA seems a reasonable size for a keyboard, so you probably won't have too hard a time finding an appropriate 40VA model, if not even a drop-in replacement. The manufacturer might have substituted a 100V primary transformer because they needed an increased secondary voltage, or it could just be an imported unit. If you can't be sure and it's been working fine for 20 years, you're probably okay just finding a 120V primary unit with a similar output rating to what you measure (given 120V, the 100V primary unit will have a proportionally increased output).
