Quite a few old amplifiers (and radios) back in the day drew power by directly rectifying the household "mains" wiring. This is an inherently unsafe practice.
Most guitars connect the bridge and strings to the ground (shield) wire on the guitar cord, essentially using the player as a "noise shield." In transformer-less amps, the Neutral wire of the mains is often used as the "ground." With a two-prong cord, Neutral and Hot can be switched (which could place the amp's ground on the Hot wire!) In other words, playing a guitar amp without an isolating transformer could be like sticking a fork in a wall outlet.
Isolation transformers limit the amount of current that can be supplied to the amp (and consequently to the guitar player) if any shock hazards arise, and eliminate possible "hot" ground issues.
In addition, we'll install a three-prong cord, so the amp has a proper earth ground. And a fuse, too. The earth ground and fuse help to maintain a sane ground reference, and protection from shorts.
And we'll incorporate the changes on a small "module," so as to change the original as little as possible. If someone is crazy enough to revert to the original setup...they can do that.
This mod works with radios, too. In fact, many of these amps were called "radio tube" amps, or "AC/DC amps"--like their radio counterparts, a transformer-less amp could be plugged directly into a DC or battery power supply without modification. A decently-sized bank of batteries were required (over 100V), but that was once commonplace.
Step 1: ZZZAAAPPPP! It's the Safety Disclaimer!
DISCHARGE THOSE POWER FILTER CAPACITORS!!!!!
Seriously. Do this EVERY TIME you work on the amp. If you don't, DO NOT complain if you loose the use of your hand. DO NOT come back and haunt me if you die....
The power 'filter' caps can store fatal amounts of electrical current, and are sometimes termed "reservoir" caps. The caps are connected near the rectifier and are part of the power supply, and aid in converting AC to DC. In fact, they are a standard component in any power supply.
If you're completely lost, and don't understand this , DON'T MODIFY YOUR AMP . You haven't enough knowledge to work on high voltage/current circuits safely...
There are several ways to discharge caps, but here's the easiest:
FIRST, UNPLUG THE AMP! (But that doesn't make it safe....)
-- Jumper the positive (+) lead of each large cap to GND for several seconds. A jumper with a built-in resistor (10K or so) will help prevent sparks here... If your jumper has a resistor, leave it connected for at least 30 seconds before you touch anything.
-- OR short the caps with a screwdriver. Lay the shaft on the chassis, then bridge to the positive (+) lead of the cap. Be sure the screwdriver handle is insulated (if it's painted, it might not be.)
This may result in a spark... Obviously, your flesh can act as a jumper also (that is NOT a challenge.)