The idea was to protect the woman from potential hurt or awkwardness, to aid her in evaluating a man whom she might not have known well at the time of his initiation, and to help ensure that the relationship was carried out honorably.
Certainly, this norm spread beyond the believing community and became more of a cultural phenomenon, but it still gels well with attempts to carry out a godly dating relationship — especially among those believers who hold a complementarian view of biblical gender roles.
It is simply God's design and assignment of equally valuable roles among spiritually equal beings. Briefly, biblical support for this position is found, among other passages, in the creation order in Genesis 2, in 1 Corinthians 11: 7-9, and Ephesians 5.
Finally, let me advocate the initiating of a relationship under some accountability structure.
I mentioned the woman's father or family because until the second half of the 20th century, that's largely how it was done.
As I mentioned, he should not do this until he is "ready" to marry.
If you're not ready to marry, you're not ready to date.