That’s a great quote, anon, and a very good question *u*

[^by @garama and the illustration is more appropriate than you think; they are probably slinging insults back and forth telepathically!]
This quote was actually Armand talking to Lestat about Gabrielle and Nicolas (Lestat had not even met Louis and Claudia yet), and how foolish it was for Lestat to expect either of them to actually want to stay with him. AS IF?! We as readers also add in the fact that it could apply to Louis and Claudia, since we already know that further down the line, yes, they will choose to leave their maker.
Armand’s point is that no matter carefully chosen, every fledgling will inevitably leave their maker. The quote actually begins with: “Each time the death and the awakening will ravage the mortal spirit,” and then he goes into more reasons why the fledglings and makers end up parting. Gabrielle left because she wanted something Lestat couldn’t give her. Nicolas left for a number of reasons, some variation of madness could be part of it. Armand speaks with the knowledge of seeing many fledglings made, and he might even feel like he would not have chosen to stay with his own maker forever, if he had had the choice.
When Armand says “And the veil will always come down between you,” that’s about the fact that the telepathic connection that can exist between vampires and mortals
(vampire!Lestat could communicate telepathically with mortal!Gabrielle before she was turned)
will be lost with the Dark Gift*
Armand had it with mortal!Daniel.
It’s a deeply intimate connection and many makers and fledglings mourn its loss.
Armand is basically saying that you can turn however many mortals into vampires you want, but due to the act of turning a new vampire, you can’t keep them as a companion. The best and most true companionship is in finding other vampires who were turned by others. There are many reasons for finding a prefabricated vampire to love, but a major one is that the telepathic connection with them will still be possible!
*At least one fic writer, @vampchronfic, has L/L finding ways to chip away and even dissolve this limitation, and it’s bound them closer together, unlike any other ship in VC.