From my understanding free radicals play a slight role in ageing.
In what ways are they so damaging, and can a restricted diet reduce production of free radicals?
Free radicals are damaging because their unpaired electrons (or not fully filled valence shell) makes them highly reactive species. They are often considered together with highly oxidizing "reactive oxygen species" (ROS) such as peroxides. They are especially problematic for cell membranes and DNA. In the latter they can react with (oxidize) heterocyclic bases.
As suggested first by Gerschmann et al. (1954), since oxidative damage to cells and cell structures is seen to accumulate with age, the corollary of this could be that aging is a result of oxidative damage. The very fact that cells have antoxidant enzymes such as superoxide dismutase (SOD) suggests that suppression of these species is important in the cell.
Many (but not all) ROS originate from mitochondria during cell metabolism. Therefore, it has been suggested that by regulating metabolism (that is, reducing the metabolic rate) the "rate of living theory" (Harman 1981) the rate of oxidative damage can be reduced. One way to do this may be calorie restriction.
This idea seems to fallen out of favour though. Another possibility is that the rate of mitochondrial ROS has nothing to do with metabolic rate per se; rather it might simply be a longevity determinant.
However, having said all this there is evidence that ROS are actually crucial in a number cellular pathways. It may be that it is the deregulation of pathways managing ROS that can contribute to aging rather than simply damage accumulation with age.
Gerschman, R., Gilbert, D. L., Nye, S. W., Dwyer, P., and Fenn, W. O. (1954). "Oxygen poisoning and x-irradiation: a mechanism in common." Science 119(3097):623-626.
Harman, D. (1981). "The aging process." Proc Natl Acad Sci U S A 78(11):7124-7128.
ROS can inflict DNA damage in cells and if this damage is persistant and cannot be repaired, the cell can undergo cellular senescence. Cellular senescence appears to play a major role in ageing (see here for example). When cells become senescent they can no longer divide and replace damaged tissue. Also, they secrete pro-inflammatory factors which can damage the microenvironment leading to disease and ageing. Senescent cells also have dysfunctional mitochondrial leading to increase ROS levels and possibly further damage.
Nice answer by Poshpaws +1. Free radicals can damage membranes (especially important for mitochondrial and endoplasmic reticulum membrane function), DNA (genes, telomeres, and mitochondrial DNA, eg), and microsomes. These are the things we consider with regard to ROS for our research on aging.
The role of diet is not clear. Smoking and alcohol may have greater effects on ROS levels than diet itself. However, people who smoke too much or drink too much generally have poorer diets (based on our (and others) epidemiological data). The role of homocysteine (Hcy) in ROS levels cannot be discounted and Hcy is influenced by genetics, diet, inflammation state in addition to other factors. In short, the evidence is not conclusive that a "restricted diet" (did you mean caloric restriction?) will certainly lower ROS levels.