Empirical descriptions of such relationships frequently use allometric functions (such as power laws), but all of the reasons why these fit the data are not understood. Some of the clearest relationships follow directly from surface-to-volume scaling necessary to maintain homeothermic organism temperature, given a certain specific metabolic rate. Kleiber's law is a famous example for respiration.
For inter-species parameters related to ecological variables such as maximal reproduction rate, attempts have been made to explain scaling within the context of dynamic energy budget theory. However, such ideas have been less successful.
The explanation of certain body size relationships differs for intra- and inter-species comparisons in the context of the Dynamic Energy Budget (DEB) theory. Young (small) organisms behave different from old (large) ones of the same species because they typically do different things (grow fast and don't reproduce). Adults of small-bodied species, however, are expected to behave similarly to adults of large-bodied species. The reason the parameters of the DEB theory vary between species may thus follow naturally from the structure of the theory.
Maximum body size itself can be written as a function of body size parameter values, because it results from the maximum food intake (which is linked to the surface area of the organism) and the maintenance costs (which are linked to the volume of the organism). Each of these parameters are thus either independent on maximum length, or proportional to maximum length. Any eco-physiological quantity that can be written as function of size parameters can for this reason also be written as function of the maximum body size.