Output transformers are rarely understood as to how they can affect the tone in an amplifier. The output transformer is the last part of tone shaping before the sound hits the speakers. Most of what it takes to create a great transformer is a lot of experience, engineering, and quality ingredients.
An output transformer functions the same as all other transformers do. As a very simplified explanation, a high voltage from the plates of the power tubes goes in to the primary side of the transformer. This is a coil of wire wrapped around a core, with a second coil next to it. This inductively changes the voltage and current to a lower voltage, and higher current that will make speakers produce sound.
These coils are usually measured in resistance. Tubes want a properly matched resistance in the output transformer. When matched properly, this allows the tubes to push at their optimum, creating more efficient power. Too high or too low will cause less power to be created, and less frequency response to be produced.
Wire has a resistance to it. Thicker and thinner wires, have different resistance values per foot. Thicker wire has a lower resistance. Thinner is higher. For instance, 28 gauge wire is 3.075 ohms (or Ω) per foot, while 32 gauge wire is 5,788 ohms per foot. So, it would take roughly 1.5 times the distance of 28 gauge wire to equal the same amount of resistance of 32 gauge. This also affects the amount of current that can be handled. Thicker wire carries more current (AKA amps). There is a balance between the sizes (and types) of wires used on the primary and secondary.
The wire is wrapped around a bobbin. Bobbins come in a variety of materials, all will slightly affect the coils that are wrapped around them (as will the distance between coils in certain applications).
The iron used in the core of the transformer matters. The core is the structure that holds everything together. Since we are dealing with electricity in a coil, we have to take some electromagnetics into account. The better the material of the core, the better the magnetic field, stability and structure.
So, how do we pick the transformer we want? With a bit of experience, a bit of education, and usually a bit of help from someone that knows a heck of a lot more than we do. The people at Mercury Magnetics have been very instrumental at turning my tonal aspirations into a physical transformer, all while helping me to understand how to ask for what I want to hear when designing a transformer. One of the most important things for us, is to properly match the load as close as possible to maximize our output and fidelity. Having a slightly close match won't ever sound as good as the proper one. If you are shopping around looking for an upgrade on an amp, ask if they are designed for your application, or if it is more of a "universal" that will work in multiple applications.