The reason for this is that shy men are more concerned with avoiding the social catastrophe of asking you out and failing than they are of getting you.
This is what prevents them from taking the initiative in the first place.
Though he knows his own limitations (as prescribed by his shyness), he also knows what a typical man is "supposed to do." So he recognizes, too, when you are doing it for him.
If they have even the slightest doubt about your intentions they will hold back.
So whatever you would normally do to indicate your interest to a confident man, double or triple it for a shy man (in proportion to his shyness): While I don't suggest that you overtly ask out a shy guy on an "official" date (this would too flagrantly undermine what he knows to be his role as described above), I do suggest that you suggest and then ask him to hang out under casual pretenses.
In general, you will need to carry the relationship up until the point that it is "official." When it has been established (or is otherwise clear) that the two of you are together, then you should start reducing your initiative and letting him take the reins.
As I said, he will probably start to do this on his own anyway, so rather than forcing it, just be cognizant of the transition.
The first thing to do when you find yourself attracted to a shy man is to question your feelings for him.