In the simplest terms (and it's a very complex subject) Il-6 (interleukin 6) levels are a way of determining if an inflammatory response is acute, or chronic.
The Il-6 peaks in acute inflammation to promote the production of the proteins that fight the inflammation, and as it drops, it would indicate that the inflammation is lessening. In the case of chronic inflammation, it levels off, and kind of switches gears from promoting an immediate response, to changing the response to a different type of white cell that has more of a controlling effect, than an attacking effect on the inflammation.
By mapping the levels based on how much they deviate from normals, and using the levels of the various white cells, it might be possible to determine how much inflammation there is, how long it has lasted, and whether it's acute or chronic.
As I said, this is a very simplified description of just one of it's functions, and your doctor may have other reasons for ordering it. It's one of those levels that doesn't mean a whole lot on it's own, it needs to be considered with several other factors like white cell count and differential, inflammatory markers, history and symptoms.