What you’ve found isn’t a separate activity. The National Catholic Committee on Scouting is an advisory committee to the Boy Scouts of America at the national level. Among other things, they have created several awards that Scouts can earn; the Boy Scouts uniform heraldry actually includes religious awards (determined by the relevant religion’s scouting committee) that can be worn on formal occasions (a cloth patch denoting “religious award” goes on the uniform for everyday wear).
Basically, your sons get involved in Scouting, just like any other boys. “A Scout is trustworthy, loyal, helpful, friendly, courteous, kind, obedient, cheerful, thrifty, brave, clean, ***and reverent.***” Just as Boy Scout summer camps make provisions for worship services, and Scouting in general expects grace at meals, etc., Scouting encourages its members to practice their religions. The boys may choose to earn awards for learning about their religion, and the award will be something they can wear to courts of honor, scouting dinners, etc.
A Cub Scout can earn the Light of Christ award (and soon the Parvuli Dei award) but really shouldn’t work on the Ad Altare Dei award or the Pope Pius XII award, because those were designed for the older scouts. In either event, he’ll have a medal for his uniform for formal occasions and a “square knot” emblem (silver square knot on a purple rectangle) to be sewn on his uniform (over the left shirt pocket) for everyday wear.
Generally, if your troop or pack isn’t sponsored by a church – or if it is, but not by your church – you simply work on the award yourself (your local council should have contact information; if not, try here). The key thing is that, even if the troop is sponsored by a church, this is not taught by the troop or pack leaders; it’s taught by the religious institution (usually not clergy; in fact, it’s often someone who has volunteered both in the church and in Scouting). There’s probably some Catholic liaison to your local council who can arrange for the boys to work on the awards if they like.