An important thing to remember is that an Hierarchical Divine Liturgy is not a "normal" Divine Liturgy with extra bits, but that what most of us consider a "normal" Liturgy is an Hierarchical Divine Liturgy with bits taken out. In other words, the Bishop vests in public because that is the original practice of the Church. Vesting in the sanctuary "before" the Liturgy is the newer development.
In the ancient Liturgy, that is best preserved in the so-called Hierarchical Liturgy, the faithful assembled first, then there was the "entrance of the celebrant"; i.e. the Bishop, followed by his vesting.
"When the president of the assembly enters, he says peace be with all of you," says John Chrysostom - confirming that this is the ancient practice. Therefore, when ialmisry says that the image is perfectly captured in the 1st/2nd century Saint Ignatius we should not be surprised at all. It is part of the Liturgy, hence my quote-marks around the word "before" above.
As to why it is done, Liza gives some reasons relating to the symbolism of the act. Even more important is the fact that for the entrance of the celebrant to have any meaning, the faithful must be assembled first to witness it. This is the ancient practice of the Church, and provides a gentle reminder about not turning up to the Liturgy late: we are simply missing out on the fullness of Orthodox worship by doing so.
I recommend Fr Alexander Schmemann's book "The Eucharist" for more on this, and in particular the chapter titled "The Sacrament of the Entrance" for more detail on the importance and reasons behind the practice of public vesting. I don't know, but you could even try Googling the chapter's title and the author to see if there are any online versions of this chapter at least.