The word "fatal" is an interesting one in this play. It's connected to another phrase that always brings it to mind: "star-crossed." In modern parlance, the word "fatal" is often used as if it means "causing death," as in the phrase, "a fatal blow." However, it doesn't actually mean that at all. It simply means something which has an important effect on the fate of someone or something. So, the "loins" of the feuding families, the Capulets and the Montagues, are not fatal because they cause Romeo and Juliet, their children, to die, but simply because they have a significant effect on the fates of both families.
Breaking down the phrase, "loins" means genitals or reproductive organs, so Shakespeare is saying that the lovers, Romeo and Juliet, have issued from the sexual organs of the Montagues and the Capulets. The implication is that the couplings of the Montagues and Capulets have had an important influence on the goings-on in Verona, probably related to the fact that they are "foes." In the next line, Shakespeare goes on to expressly state that the lives of Romeo and Juliet, the lovers, stem from these loins, which again simply means that they are the descendants of two houses which are at war. The phrase overall is simply a clear statement that Romeo and Juliet, the hero and heroine of the play, seem to have had their lives defined by some element of fate: because they have been born from two warring houses, their coming together has had a significant effect on their own lives and also the world around them.