Other translations present that convoluted "that also which he thinketh he hath" as "even what he thinks he has," in case that tripped you up, too; sometimes the Douay-Rhiems is harder to untangle than just reading the passage straight from the Vulgate!
The seed that fell of the path: it just sat there on top of the path, and so it was taken away by the birds. The seed that fell on the rock: it sprang right up, but not being nourished, died right out again. The seed that fell among thorns: it grew, but not being made a priority, was slowly choked out. All thought they had the seed, the faith, but because they didn't actively take possession of it, seek to develop it, give it priority in allocation of resources, they didn't truly have it.
The concept of "to whosoever hath [faith], to him shall be given" is illustrated all over the Gospels, but we get two that I particularly remember in this chapter: the woman who touches Christ's cloak, and the daughter of Jairus. This woman has been hemorrhaging for 12 years, has spent all her money on doctors who haven't been able to cure her, and believing Christ can cure her, she touches the hem of his cloak and is immediately healed. "Thy faith has made thee whole," Christ tells her. She had faith, and to her was given healing.
Jairus is the "ruler of the synagogue" according to the Douay-Rheims, a pretty direct translation of the "princeps synagogæ" from the Vulgate, not that that actually tells me a whole lot about his role in the Hebrew culture because I'm not familiar with the relationship between the synagogues and the Temple. But, a title like that, and the fact that a few verses later he's referred to as Master ("Præceptor"), makes him sound fairly important. Already by this point in Luke's Gospel, many of the important people connected to the Temple want Jesus dead. But Jairus comes before Jesus, prostrating himself, begging Him to save his daughter from death. And when Jairus is informed that his daughter has died, Christ tells him to "fear not; believe only, and she shall be safe." Jairus believes, he has faith, and his daughter is returned to him.
One thing we can't try to take from this is that having faith means that all good material things will come to us, or that bad things happening despite our prayers means we don't really have faith or our faith wasn't good enough. Nope. Very frequently throughout the Gospels, Christ's initial response to someone who is seeking physical healing is "Your sins are forgiven." He heals our spiritual ailments first, and then, if it would be the best thing for our souls, He heals physical or otherwise material ailments. Sometimes, avoiding or removing a form of suffering would be bad for us spiritually. Sometimes it's better for us to be given the strength to bear something for His sake than to have it instantly made better. And He wants what's actually best for us, not just what we think is best or easiest.