The prophecy seems to be set at the advent of Christ, because it says in the prior verses:
Isaiah 35:4-6 (DRB)
Say to the fainthearted: Take courage, and fear not: behold your God will bring the revenge of recompense: God himself will come and will save you. 5 Then shall the eyes of the blind be opened, and the ears of the deaf shall be unstopped. 6 Then shall the lame man leap as a hart, and the tongue of the dumb shall be free: for waters are broken out in the desert, and streams in the wilderness
"The Way" was what Christians were known as well as "Christians."
Who persecuted this Way [Christians] unto death, binding and delivering into prisons both men and women.
But this I confess to thee, that according to the Way, which they call a heresy, so do I serve the Father and my God, believing all things which are written in the law and the prophets:
This language of following a Way is even used in an extremely early Christian catechetical work called the Didache, which speaks of "the Way of Life," and "the Way of Death:
Chapter 1. There are two ways, one of life and one of death; but a great difference between the two ways. The way of life, then, is this . . .
This way of separation from 'the world' is what makes it "holy." And because it requires holiness, and leads thereto.
Follow peace with all men, and holiness: without which no man shall see God.
Of course it's a "holy way" because ultimately they are following Jesus Christ as "the Way." That's why it is called holy and uses "the unclean shall not" as a parallelism/contrast. Jesus is the very Way itself, which leads to God the Father:
Thomas saith to him: Lord, we know not whither thou goest; and how can we know the way? 6 Jesus saith to him: I am the way, and the truth, and the life. No man cometh to the Father, but by me.
"...but they shall walk there that shall be delivered." That is, only those who are of the elect will persevere in this Way after which they will have reached their goal in heaven.
There shall not enter into it any thing defiled, or that worketh abomination or maketh a lie, but they that are written in the book of life of the Lamb.