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When suggesting nitration of an aromatic compound in the synthesis of some organic molecule, it was raised that this route should be avoided as to prevent things from going "ka boom."

An explanation was not forthcoming.

So, why do nitro groups tend to make organic molecules explosive? Is it because the $\ce{NO2}$ group really "wants" to be $\ce{N2}$ since a) diatomic nitrogen has a super high bond strength and b) diatomic nitrogen is a gas usually and therefore conversion to diatomic nitrogen would be entropically favorable? Is it because carbon's most oxidized state is also a gas and the $\ce{C=O}$ bond isn't a wimp either - hence the reason many organic compounds are flammable?

Also, what happens first in an explosion? Does carbon become oxidized first, which then provides the activation energy necessary to decompose the nitro group? I ask this because organic carbon compounds by themselves which are flammable - i.e. toluene - aren't considered explosive, but trinitrotoluene is an explosive.

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Explosives chemistry is a rather complex topic. I've heard that this book is a good source of information about it (I haven't read it).

In a nutshell, your intuition about the nitro group is accurate. Formation of $\ce{N2}$ is highly energetically favourable. To get an explosive, what we need is a rapid reaction that produces a lot of heat and gas to both cause the obvious effects of an explosion and to propagate the reaction to other molecules of explosive. The rapid part is what separates something like TNT from toluene. The combustion of toluene is also energetically favourable, but in TNT, to create $\ce{N2}$ is not dependant on mass transport of any other species and can thus happen very rapidly, whereas the combustion of toluene is limited by how quickly oxygen is transported to it. If one were to vaporize toluene in the correct concentration in air, this transport problem goes away and an explosion can occur. TNT also has an advantage in this regard because the nitro groups provide a source of oxygen to react with the carbon and nitrogen remaining (not enough for all of it, but it helps). Explosives are often mixed with fuels or oxidizing agents to produce a more oxygen-balanced mixture for a more efficient explosion.

As for what happens first in an explosion, many different reactions can occur during an explosion, but it may be helpful to consider what it takes to actually detonate TNT and think about the timescale of the reactions. TNT is a solid at room temperature and has a flash point of 163 °C making it difficult to even ignite, and while it will burn in a fire, there is no risk of explosion. For an explosion to occur, enough gas and heat has to be produced to propagate the reaction through the bulk of the material. In practice this is done using a much more sensitive explosive (other explosives like lead azide or nitroglycerine are unstable enough to be set off by heat or pressure) to produce a small shockwave that provides the activation energy to initiate a reaction as it travels through the explosive which then sustains the shockwave through the rest of the material. In a normal explosive (not a fuel-air explosive or the like), the reactions that contribute to the bulk of the explosion are limited to what the explosive is made of because the speed of the explosion is too fast for air to play much of a role initially.

In the case of TNT, the experimentally-measured time it takes for a shockwave to pass through is 100–200 fs (no idea how one measures that), so any oxidation of the carbons seem unlikely to contribute much to the the initial explosion, given the only readily available source of oxygen is from the nitro groups which must presumably decompose first. This group proposed a few decomposition pathways for TNT, including homolytic cleavage of the $\ce{C-NO2}$ bond, rearrangement from $\ce{C-NO2}$ to $\ce{C-ONO}$ followed by homolytic $\ce{O-NO}$ cleavage, and $\ce{C-H}$ attack from an adjacent nitro group to the methyl ring substituent, but they found that only the first was fast enough to occur during detonation, the others possible only for lower temperature thermal decomposition. This initial decomposition step is the only thing fast enough to contribute to the shockwave that sets off the rest of the TNT, while the reactions that produce the final products occur (relatively) long after the initial blast has initiated the rest of the explosive.

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What may make things go kaboom without true explosion

First of all, whenever you have a mixture of an oxidizer and a fuel, whatever nature, there is always risk of fast combustion. Common gunpowder is a mixture of solids, and it can go kaboom when confined. Since nitrogroups are oxidizing, while the rest of the molecule is usually reducing, many nitrocompounds are capable of combustion without external oxidizer. The process may proceed without true explosion, for example, TNT in small portions may be safely ignited. It burns with bright yellow flame producing a lot of black thick smoke. Anyway, to produce a kaboom, the compound or mixture must have a way of decomposition producing a lot of energy.

Self-sustaining thermal decomposition, say of hydrogen peroxide, may produce large volumes of gases, and in case the reaction mixture is confined, the build-up of pressure may eventually lead for the reaction container to... fragment violently.

Detonation wave

A true detonation is described as a process, where instead of burning front, proceeding with subsonic speed, a supersonic detonation front is observed. The speed the front travels with may be above 10 km/s in some solids and over 3 km/s in gases. This implies that the process does not involve diffusion of active species or thermal energy, that travel usually with subsonic speed. So decomposition in detonation wave is pressure-induced, either from adiabatic heating or from direct mechanical stress. To achieve this, the compound must be able of exothermic decomposition, producing enough energy to sustain the pressure wave.

Nitrogroup role

Commonly utilized nitro-explosives are relatively stable and often require a significant primary explosive charge to induce detonation wave. Nitrogroups, despite having positive energy of formation, are relatively durable groups, with high barrier or bond dissociation energies, double so for nitroaromatics. This means, that in detonation wave decomposition of nitrogroups mostly consumes energy, while formation of other molecules, say, water and carbon monooxide, produces energy. So, the role of nitrogroup is mostly to provide a readily available oxygen for oxidation of the rest of the molecule.

Other cases

However, there is a lot of explosives (most commonly with high amount of nitrogen), that have comparatively low decomposition energy, but are extremely sensitive. Probably, ones of the most infamous would be heavy metal azides, like lead azide. There in addition to readily available exothermic decomposition route, usually some weak bonds are present in the molecule, allowing easy thermal fragmentation, following by rearrangement.

For example, infamous diazomethane have C-N bond energy around 172 kJ/mol, while energy of $\ce{CH2=CH2}$ bond is 611 kJ/mol, meaning that the process produces roughly twice more energy than consumes - not a lot, but the very low initial energy investment makes diazomethane to explode from a funny look.

TL;DR.

nitrogroups mostly serve as a readily available source of oxygen to oxidize the rest of the molecule

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