Snowbum's BMW Motorcycle Repair & Information Website

The simplest way to describe it, and this is my personal explanation, is that if you are at 'some' amount of RPM below the yellow (caution) line on the tachometer, and increase the throttle, expecting more power and thus more speed, the result is no or hardly any increase of RPM, no or hardly any increase in power, no or hardly any increase in speed.  As lugging amount increases from too heavy throttle for the RPM, the engine may even exhibit a thudding sound.   The result of lugging is that the combustion heat generated by the engine increases, particularly at the exhaust valve; but, can also be at the piston top, top ring(s), etc.  Lugging is not good for some bearings, and other things.   Lugging at moderately low to moderate RPM can be quite bad for the engine, for a stock Airhead, perhaps around 3000-6000 RPM is the area that lugging is more likely to be damaging.  NOTE, however, that lugging is possible at almost any RPM.  Note also, that there is a special type of lugging that is not generally considered damaging.  This is usually NOT described as lugging. This type of non-lugging is typically mild, done at typically perhaps 2000 to 3500 RPM, and you are likely on a flat road, holding a SMALL amount of throttle, and likely in a gear that does not require much throttle. 

At this point, I recommend you review the following two articles, for photos, etc:
https://bmwmotorcycletech.info/excessive.htm
https://bmwmotorcycletech.info/setvalves.htm

REAL lugging has specific causes:

RPM too low, resulting in too low a power to enable an RPM increase.   Cure:  Drop down a gear or two.
Carburetion is too lean.  Cure:  enrichen the carburetor.  This is, or can be, complicated, as the type of carburetor (pure slide or CV)(or, fuel injection settings/tuning, which does not apply to a stock Airhead) is involved, as is the jetting, needle type, needle clip setting, fuel height in the fuel bowl, etc., is involved.  Some of these things affect each other. 
Wrong fuel: INsufficient octane value of the fuel, and whether or not the fuel contains such as ethanol.
Engine design more susceptible:  This applies to the R65 and R45 that have thinner valve guides, and less ability to remove heat from the exhaust valves.   Note that longer stroke engines usually don't take to lugging situations in exactly the same manner as short stroke engines.

Serious lugging can be very bad for your engine:
The exhaust valves and piston tops, perhaps top ring(s) too, can overheat, and the parts melt or otherwise fly apart, with metal pieces...some large...some small....getting into the engine oiling system, ETC.  See the two hyperlinked articles near the top of this article, one has pictures of such disasters.