I think I might interpret your question as asking, not just why don't mammals have more than four limbs, but why arthropods have more variety. Insects have six, but others have eight, ten, or more.
Partly there are just many more species of arthropods. 80% of animal species are some sort of arthropod, and some lineages of arthropods are distantly related to each other because arthropods as a group have existed since the Cambrian period. With so many more species, it's no surprise that they've managed to tack on some extra variety.
Although it might seem that insects and spiders are more closely related than, say, humans and fish, the reverse is actually true. In fact, insects split off from other arthropods at approximately the same time that fish were diversifying into the three main lineages that would eventually become cartilaginous fish (sharks & rays), ray-finned ("regular") fish, and lobe-finned fish (coelacanths and lungfish). All of these things happened at approximately the same time, geologically speaking, at around 400 million years ago. Some ancient lungfish crawled up onto land a short while after, giving rise to land animals. So in light of this, it's not really surprising that arthropods would have so much diversity in body plan - you don't look much like a leafy sea dragon (I hope).
But perhaps another way of looking at it is in the difficulty involved in adapting the body plans of these different types of animals. Arthropods have segmented bodies. 445
In some arthropods, especially the myriapoda (centipedes, millipedes and so on), there's a pretty straightforward arrangement of one segment to one pair of legs (excluding the head). Vertebrates have a segmented body plan too, but our segments aren't quite so well-separated. In vertebrates, a "segment" is approximately equal to one vertebra. So since vertebrates aren't in the habit of attaching a pair of legs to each vertebra, it's harder to grow more legs. Instead, vertebrates have internal skeletons, with the familiar hips and shoulders in land animals. Vertebrates do have good luck adapting the internal structure when they need to. Most use the traditional four-leg layout, but humans, birds, whales, and snakes have all made pretty significant changes to it. But internally, it's still the familiar hips and shoulders - even snakes still have the necessary bones. Given another few hundred million years, they might disappear entirely, but it seems like the basic bone structure is tough to change. Who knows, in a hundred million years, the far future descendants of snakes might end up with legs again, and grow six or eight instead of four!
So, overall, I'd say that the two main reasons are that the arthropod body plan is just a little more flexible due to its higher degree of modularity, and that different kinds of arthropods are actually more distantly related than you might expect. 335