Short answer. They inherit in their own right. Article 969 says that when the nearest relative, or all the nearest relatives called to succeed, repudiate the inheritance, those of the next degree inherit on their own, not by representation. You cannot represent someone who has repudiated, so the next degree simply steps up and takes directly.
What the law says
those of the following degree shall inherit in their own right and cannot represent the person or persons repudiating the inheritance
Civil Code, Article 969 — Repudiation by All in a Degree. Read the full provision →
Repudiation clears the nearest degree
Inheritance moves outward by degrees of relationship, and the nearest relatives are called first. But being called is not the same as accepting. Article 969 addresses what happens when the nearest relative renounces: if there is only one and he repudiates, or if there are several and all of them repudiate, that degree drops out of the picture. The estate does not sit unclaimed; it passes to the relatives of the next degree, who now take their turn to inherit.
The next degree inherits in its own right
When the nearer relatives step aside by repudiating, the more distant relatives who follow do not inherit as substitutes standing in the renouncers' shoes. The article says those of the following degree shall inherit in their own right and cannot represent the person or persons repudiating the inheritance. They inherit directly, as the persons now nearest in line, and they divide the estate among themselves according to their own relationship to the deceased, not according to the share the repudiating relatives would have taken.
You cannot represent someone who repudiated
This is the key limit. Representation, the mechanism that lets descendants step into the place of a relative who cannot inherit, is available when that relative has died or is disqualified as unworthy. It is not available where the relative simply chose to renounce. A person who repudiates is treated as having declined the inheritance for himself, and his own children cannot represent him to claim it. The line of representation is broken by a free repudiation.
What it means in practice
The practical effect is that renouncing an inheritance does not preserve it for your branch of the family through representation; it releases the share to whoever the law calls next in their own right. Someone deciding whether to repudiate should understand that they are not passing their portion down to their own descendants by doing so. If the aim is to benefit the next generation, repudiation is usually the wrong tool, because the estate will pass by the law's ordinary order rather than by representation of the one who renounced.