As tiny as the signal is from a MC cartridge, it is a, relatively speaking, high current/low voltage source. The phono amplifier is designed as a voltage amplifier. The step up transformer converts high current/low voltage to high voltage/low current, which means that the amplifier has more voltage to work with and amplify. As the name suggests, it transforms current to voltage so it is not actually amplifying the signal--there is no need for an additional source of power and active devices to amplify the signal.
Is it better to employ active amplification instead of current to voltage transformation as the first stage? Opinions obviously vary on this. Some people eschew use of transformers while others prefer it over an additional stage of active amplification. With a lot of tube phono stages, the use of a transformer for the first stage is preferred because active tube amplification is somewhat noisy, and eliminating the first stage by using the "free" conversion of a transformer means less noise. I suspect that some people, myself included, like the sound of transformers (i.e., euphonic coloration).