The impulse response is in essence a recording of what it would sound like in the room if you played an extremely loud, extremely short click - something like the crack of a pistol shot, but shorter. The reason for measuring the impulse response (by more subtle means than firing a gun in the room) is that it completely characterises the behaviour of the system consisting of the speaker(s) that were measured and the room they are in. An important property of an impulse, not intuitively obvious, is that it if you break it up into individual sine waves you find that it contains all frequencies at the same amplitude. Strange but true.
This means that you can work out a system's frequency response by working out the frequency components that make up its impulse response. REW does this by Fourier Transforming the impulse reponse, which in essence breaks it up into its individual frequency components. The plot of the magnitude of each of those frequency components is the system's frequency response.
When an impulse response is measured by means of a logarithmically swept sine wave, the room's linear response is conveniently separated from its non-linear response. The portion of the response you see before time 0 is actually due to the system's distortion - if you look closely, you may see that there are small, horizontally compressed copies of the main impulse response there - each of those copies is due to a distortion harmonic, first the 2nd harmonic, then the third, then the fourth etc. as time gets more negative. The period after time 0 is the system's response without the distortion.
In a perfect system of infinite bandwidth with totally absorbent boundaries, the impulse response would look like a single spike at time 0 and nothing anywhere else - the closest you get to that is measuring the soundcard's loopback response. In a real system, finite bandwidth spreads out the response (dramatically so when measuring a subwoofer as its bandwidth is very limited). Reflections from the room's boundaries add to the initial response at times that correspond to how much further they had to travel to reach the microphone - for example, if the microphone were 10 feet from the speaker and a sound reflection from a wall had to travel 15 feet to reach the microphone, that reflection would contribute a spike (smeared out depending on the nature of the reflection) about 5ms after the initial peak, because sound takes about 5ms to travel that extra 5 feet.
When measuring full range responses from loudspeakers (rather than subwoofer responses) the reflections are easier to spot as the higher bandwidth of the full range system keeps the spike of the impulse (and the reflections) quite narrow, but you need to zoom in on the time axis to see them. They are easier to spot with a linear Y axis (set to %FS instead of dBFS) and also show up more readily if you select the "Show Full Range Energy-Time Curve" option.
You can "clean up" the impulse response by placing absorption at the primary reflection points, which you can find using the mirror trick - have someone hold a mirror against the wall (or floor or ceiling) while you sit in your listening position and move it until you can see each of your speakers, the positions where your speakers are visible are the primary reflection points. Absorbers at these points reduces the spikes after the main peak which are due to the reflections, using larger areas of absorber will also provide a general reduction in the "liveliness" of your room.
The main contributors to the slow decay of the impulse response in full range measurements are low frequency modal resonances, addressing these (by EQ and/or room treatments) will correspondingly speed up the decay of the impulse response.
HTH,
John