1. Because these compounds gain so much stabilization through their conjugated pi systems (i.e. they are resonance stabilized), it may be easier for the body to synthesize aromatic compounds over other molecules of similar complexity. This stability may also increase their halflives within the organism, making them less prone to degradation (important, for example, for DNA).
2. In binding reactions between two biomolecules, there are two important factors influencing the strength of binding: the enthalpy change (ΔH) of the interaction and the entropy change (ΔS) of the interaction. While many think of the enthalpy of the interaction (how strong are the intermolecular bonds between two interacting molecules), the entropy of the interaction is just as important. Molecules that are conformationally flexible in the free state lose a considerable amount of energy upon binding to their target as this binding usually locks the molecule into a single conformation. Aromatic rings are nice in this regard because they are already conformationally locked and will not lose as much entropy upon binding.
3. As Yanick mentioned, aromatic rings can aid intermolecular interactions on the enthalpy side as well. A number of fairly strong intermolecular interactions involve aromatic rings, including pi-pi stacking interactions and cation-pi interactions.
4. The conjugated pi systems in aromatic rings are great for helping to delocalize charges and radicals. As Yanick mentioned, this property allows the aromatic rings to facilitate a number of chemical reactions.