Here is an example to discuss specifics against:
> plot(stl(nottem, "per"))
So on the upper panel, we might consider the bar as 1 unit of variation. The bar on the seasonal panel is only slightly larger than that on the data panel, indicating that the seasonal signal is large relative to the variation in the data. In other words, if we shrunk the seasonal panel such that the box became the same size as that in the data panel, the range of variation on the shrunk seasonal panel would be similar to but slightly smaller than that on the data panel.
Now consider the trend panel; the grey box is now much larger than either of the ones on the data or seasonal panel, indicating the variation attributed to the trend is much smaller than the seasonal component and consequently only a small part of the variation in the data series. The variation attributed to the trend is considerably smaller than the stochastic component (the remainders). As such, we can deduce that these data do not exhibit a trend.
Now look at another example:
> plot(stl(co2, "per"))
If we look at the relative sizes of the bars on this plot, we note that the trend dominates the data series and consequently the grey bars are of similar size. Of next greatest importance is variation at the seasonal scale, although variation at this scale is a much smaller component of the variation exhibited in the original data. The residuals (remainder) represent only small stochastic fluctuations as the grey bar is very large relative to the other panels.
So the general idea is that if you scaled all the panels such that the grey bars were all the same size, you would be able to determine the relative magnitude of the variations in each of the components and how much of the variation in the original data they contained. But because the plot draws each component on it's own scale, we need the bars to give us a relative scale for comparison.
Does this help any?