The respiratory system and exoskeleton are not actually as strict of constraints as most people have been led to believe. For example, it's not actually the use of trachea in and of themselves that limits access to oxygen--it's reliance on passive diffusion. Improving the circulatory system and adding pumping to the function of the trachea (which some insects already have!) take care of that problem just fine.
Exoskeletons are more of a problem, but not in the way that is most obvious. Yes, they can be heavy, and that limits the growth of terrestrial arthropods somewhat--but in areas where there is little or no competition from tetrapods, arthropods can get really big already--and that's without even considering extinct clades like the famous carboniferous-era meganeurans. Consider, for example, the coconut crab, which can be up to 1 meter long!
The biggest problem with exoskeletons is the need to shed them as the animal grows. Molting gets harder as the organism gets larger, and it can become a very stressful and potentially deadly experience. Lobsters, for example, are sort-of functionally immortal, in that they show little to no signs of senescence, and just keep getting bigger as they age--but if they don't get eaten first, they will eventually die from the stress of necessary molting as they just keep growing. In fact, there are plenty of insects that just don't ever molt in adult form, because it's just not worth it--it wouldn't extend their lives enough to justify the risk. On the other hand, though, molting does come with advantages--it gives you a way to heal from massive injuries. Every molt is basically a full morphological reset.
Exoskeletons are thus not as good as endoskeletons for growing large, but they aren't inherently bad either. In competition with tetrapods, large arthropods tend to lose out--but your whole scenario is about eliminating that competition! Thus, I would expect to see quite a variety of quite large insects with little to no major development in their externally-obvious features.
To improve oxygen delivery, you'd want a more developed circulatory system, with more closed components; it need not evolve into a completely closed system, but it would be helpful to have better specialization of haemolymph vessels dedicated to picking up oxygen from the trachea and distributing it to the rest of the body. Additionally, you could see the evolution of "diaphragm" muscles designed to flex the body wall, thus changing internal volume to actively pump air in and out of the trachea; this sort of development has already happened once with the development of insect wings from the body wall.
In order to deal with the exoskeleton, you can imagine basic structural changes to make them more efficient at large size, like thinning out the skeletal wall on one side of a limb and thickening it on the other, with reinforcing spars extending into the interior of the limb, so it acts structurally more like an endoskeleton with muscles clumped on one side. None of that would be externally visible, though. The jackpot would be evolving a method of digesting and remodelling chitin after it has been laid down, thus eliminating the need for molting as the skeleton could then grow with the rest of the creature. That seems like the least likely option, but it would also not be externally visible.
Another way to sidestep the molting issue is to just not molt that much. You can have large eggs that produce large initial larvae, that grow even larger and metamorphose directly into a pretty-darn-big adult, which then does not have to molt as many times (or perhaps at all) to attain its final size. Adult molting may be retained in some clades as a response to serious injury, though--sure, the molt might kill you, but if you don't molt, you'll die anyway, so might as well roll the dice! If you win, you get a full reset and more years to reproduce.
As for what changes would actually be visible in adults--what they would look like--it's all basically square-cube stuff. You either get creatures that all stay very low-slung to the ground, distributing their weight over a large area, or you take the basic design of grasshopper back legs and apply that to all the weight-bearing limbs--i.e., move them closer to the body and more up-and-down, rather than splayed out to the side--to permit taller-standing creatures, and you increase the relative cross-sectional size of the limbs compared to the body as the whole creature increases in mass. Thus, a meter-scale mantid, for example, would probably have four enormous beefy grasshopper-like legs, with the front two oriented nearly straight up-and-down (and the back ones bent solely so that the positions of the feet form a large and more stable box than the close-together thorax attachment points would otherwise permit), rather than the wispy legs and wide-box stance typical of present-day mantids, while the non-weight-bearing forelimbs could remain essentially unchanged in relative scale.
