Relation between transformer core material and application frequency
It is difficult to explain in an intuitive way. But I like to think of it in terms of primary inductance. When you apply a voltage to the primary, even if the secondary is open-circuit, a certain amount of current flows in the primary, because of the primary inductance. In other words, when the secondary is open circuit, the primary just looks like an inductor. You want that current to be small, which means that the inductance needs to be large.

As it turns out, for 50 or 60 Hz, it is not really practical to do an air core, because the inductance will be too low for practical core sizes. But when a low reluctance core is used, the primary inductance is much higher, and the transformer size is more manageable.

The cross section area of the core and the number of turns are usually fine tuned so that the magnetic field strength in the core is within a range that is suitable for the transformer core material.