"lean burn" refers to the air-fuel ratio of as much as 22:1 that the 5spd Insight can use. Most cars run at about 15:1. Lean burn is what allows the 5spd to cruise while consuming very little gas, even on the highway. http://www.insightcentral.net/encyclope ... ngine.html
and the catalytic converter will work by doing occasional rich-burn "purges" which you will see as increased fuel consumption on the MPG display and feel as a slight boost in power: http://www.insightcentral.net/KB/faq-no ... ngineSurge
Hope this helps!
EDIT - Note that the CVT Insight does not use lean-burn, so most of what I said above is only applicable to the 5spd. However you may still find the first two links useful in general.
Is there anything on the dashboard to indicate that the engine has entered lean burn mode? Or do you have to be aware enough to notice that your mileage has suddenly increased with the shift?
The FCD will suddenly jump, then you know your in lean burn. Try holding the FCD at about 95, 100 mpg while going about 60 down the highway and watch the mileage go up.