If air is a bad conductor, how does fire heat up a room?

Air may be a bad conductor, but it's not that bad a transporter. What I mean, is that the room is heated by a process called convection, and not conduction.

So basically, fire heats up its neighboring air, which gets lighter and rises up and is eventually displaced by cold air which again gets heated and rises, and the process continues!

There are three mechanisms at play: conduction, convection and radiation. Radiation is the most immediate. Your environment irradiates you with black body radiation at room temperature (assuming that you are in a room at 20 C/ 293 K). As soon as your fire burns it emits black body radiation at a temperature of a 600 C (900 K), that is mostly in the infrared. The power emitted by a black body is proportional to $T^4$ (Kelvin), according to the Stefan-Boltzmann law, and this radiation is quite intense. Also the air will be heated by your fire and hot air will reach you by convection. The least important effect is air conduction.

Consider also that fire, and any heated surface, emits infrared light, which you feel instantaneously. Thus, if you turn on a heating plate, you may instantly feel a change in temperature.

Air is a terrible heat conductor, but it is also dynamic. Warm air simply gets out of the way to make space for cold air, which increases the rate of heating. Because the hotter something is, the slower it heats further, as the heat gradient becomes smaller. Thus, if the hot air moves out of the way, making space for cold air, that heats faster, it increases the perceived heating rate.

Probably the most important heat transfer mechanism in the development and spread of a fire in room is radiation which can raise the temperature of the materials in the room to their ignition point. The main role of the air in the room is to provide oxygen to enable ignition and self-sustained combustion.

Hope this helps.

Most of the heat is transported by air flow. Often this is just convection*, but if you use a wood stove for heat (as I do), you'll find that it heats much more effectively if you use a blower fan to help move heated air away from the stove and into more distant areas.

A second significant transport is radiation. That's one reason (the other being aesthetics) why many wood stoves have large glass windows.

*Note that most insulation - glass wool, foam, even the fluffy fabric of a sweater - is designed to trap air, and thus minimize convective heat transport.

Preamble
In physics, when one says that something is big/small or good/bad, these terms are never meant in absolute sense, but relative to something else. E.g., we cannot say whether 1 meter is a big or small distance: it is small when we talk about stars and huge when we discuss atoms.

Is air a bad heat conductor
I assume that conductor in the OP means heat conductor. Air is a bad heat conductor, if you compare its heat conduction with that of a piece of a metal or a single layer of glass. Thus, windows in northern countries are often made of a double glass layers, separated by air, so as to preserve the heat in the room from leaking outside. Still, the windows remain the main route for the heat loss, unlike the walls filled by better isolating materials, i.e. the materials that conduct heat even worse than air. Why do we not fill windows with the materials that are better isolators than air? - because we want them to be transparent.

Similarly, although air may be a relatively bad heat conductor, we do not have much choice when heating our rooms than to allow it to be filled with air.

Convection and radiation
It has been pointed out in the other answers that the heat exchange between the fire and surroundings is mediated not only by air, but also through the radiative heat transfer. Let me characterize both in terms of easily observed physical phenomena.

Convection
Firstly, convection is not the same as heat conduction: conduction occurs through an essentially static material, whereas convection is displacement of volumes of already hot air to another place. The flow of air is always present in ventilated human habitats - which is what we encounter in everyday life. Even if no particular effort is made to ventilate a room, the warmer air typically accumulates at the top of the room and eventually leaves it, whereas a cold air enters along the floor. This could be tested in an easy experiment, by placing a candle in the door opening at different heights from the floor.

When a room is heated by the stove, the air around the stove becomes hot, raises up, and spreads along the ceiling or penetrates at the higher floors. Since air is a relatively bad heat conductor, it may take quite a bit of time until the heat propagates down, and the floor in rooms heated by a fireplace often remains cold. Note also, that maintaining a fire in a stove requires quite a bit of convection to supply oxygen for combustion, and stoves are explicitly designed for this purpose.

Radiation
Radiation plays an important role in heat transfer between fire and its surroundings. If we sit near the fire we feel heat even though we are not in the stream of the hot air rising upwards. Wet clothes placed in front of a fire dry rather quickly. These effects take place, even if we reduce the contribution of convection to a minimum - e.g., when sitting in front of the camp fire, where hot air escapes completely.

Note also that bad heat conductors are also characterized by higher heat capacity. Thus, a metallic stove may heat very quickly, but would not keep any heat after the fire is extinguished. On the other hand, a brick fireplace may take a few hours to heat up (and a few hours to heat the room, since air heat conduction is slow, and there is no much radiation coming out of a closed fireplace). However, once hot, it may keep the room warm throughout the night (it is necessary to note that it similarly heats the room via convection and radiative transfer in the infrared range - it is this infrared light that is perceived as heat when we are next to a fireplace, but do not directly see the fire).

Remark: It was pointed in the comments, the windows are nowadays filled not necessarily with air, but with other transparent materials having poor heat conductance (including vacuum). While this is an interesting piece of information, the point I was making is that the windows are filled with air not just because it is a good insulator - the need for transparency restricts the range of materials that we can use, and air is not the worst choice.

Turbulent vs molecular transport

Molecular diffusion of scalar quantities by fluids is often very slow. If you open a jar with a stinking substance, you won't e able to small from some distance due to molecular diffusion, but due to turbulent transfer. The fluid in a typical room is moving chaotically and carrying scalar quantities with it.

For heat (internal energy) or temperature, we call the molecular process conduction (and we use a different coefficient and the Prandtl number instead of the Schmidt number, and ...), but it really works the same way. The turbulent Prandtl and Schmidt numbers (the ratios of turbulent viscosity to turbulent diffusivities of heat and passive scalars) will actually be almost the same so the efficiency of diffusing heat and passive scalar quantities by turbulent motions is the same.

Convection

Convection is an ambiguous or overloaded word. Often, depending on the subject of sturdy, it is used for what other would call advection but in many engineering disciplines using "convection" with multiple meanings seems to be preferred. The term "advection" has the advantage of being non-ambiguous. The turbulent motions of air in the room with a fire or with an open jar with a stinking substance advect the heat or scalar concentrations with it.

But convection has also a different meaning. convection is the turbulent motion generated by thermal stratification that is unstable, typically by heating lower parts of the room, but can also be started by cooling the upper parts of the room. Convection causes large organized turbulent motions that are very effective in mixing scalar quantities in the room.

Convection will start when the Rayleigh number is large enough. For a fixed room with air it means when the temperature gradient becomes large enough. The temperature difference necessary to start convection is quite small and any room heating will start it. That's why radiators are able to heat a room. They would not be able to do it just by molecular heat conduction.

The term "convection" also appears in the terms "forced convection" vs. "free convection" often discussed in engineering, that are related to the present problem, but I will not touch thme.

Radiation

Any solid body (or even the fire) with nonzero emissivity will emit radiation based on the Stefan-Boltzmann law. The intensity is relative to the fourth power of temperature (~T^4).

Therefore, objects in direct view of the fire will be heated by radiation. That radiation happens mostly in the visible and the infra red-part of the spectrum (depending on the temperature of the object).

Let's flip it around: why doesn't it the heat the room up more?

For a good visual, imagine you're out in just-above-freezing temperatures. Which would you rather do:

• not have a glove on your hand, exposed to the air
• put your hand into nearly freezing water
• put your hand on a flagpole and hold it there

... the first! Having some skin exposed to cold air isn't nearly as bad as cold water, or cold steel. Because the air doesn't conduct nearly as much heat.

Now think about that fire. It's something several hundred degrees warmer than your body is equipped to handle... yet you're able to stand a meter away from it! When you think about it, the air's doing an awfully good job of insulating you away from that heat (and most of the heat you feeling isn't even coming from conduction, but from radiation).

Now imagine that the large fire was build atop a thick metal plate. Do you think you'd be able to stand on the metal plate that close without burning your feet?