I see lots of posts, so if this has been said already then I apologize, I looked at them all, but I just scanned them quickly, and I didn't see this info said.
I think the question at hand (why higher bypass engines are more efficient) can be answered using a thermodynamic approach. Basically in a turbine engine we have a simple cycle taking place. First there is a compression of air that causes the air to heat up as well as increase in pressure (as per the ideal gas law). This is followed by a constant pressure addition of heat (in the combustion chambers), then an expansion of the hot air generates power in the turbine, and the resulting warm air is sent out the back of the engine (I say warm, as it's cooled significantly from the hot section by the air expanding in the turbine). This is the Brayton cycle. I think most people here probably understand this process quite well. Now to bring the question of efficiency into this problem. There are ways to calculate efficiency in theory, but I won't go into figures. Basically the theoretical efficiency of the cycle is increased if the temperature in the hot section is increased. Now when you have a turbine operating like this there is a significant amount of excess energy. One approach is to simply allow it to go out the back of the engine (ie- turbojet) and provide forward thrust. However, if you're looking at a turbofan engine, if you increase the hot section temperature, you have more excess power available to drive a larger bypass ratio fan. So basically I think this answers the question: the combustion section temperature dictates efficiency of the turbine to a good extent, and the hotter the temperature the larger the fan it can power. In actuality it is the heat resistance of the metal used in the hot sections of the engine that determine how high the temperature can go, and thusly the efficiency.
Just an anecdote: turbine inlet temperatures were at about 1000F (540C) during the 1940's, and have increased to roughly 2600F (1425C) today, obviously this is a good part of the factor in improving engine efficiency in the last 60 years since Sir Frank Whittle.
It is my understanding that this also explains the poor efficiency of many Russian built engines. I have been told, anyways, (not by a prof, but by someone who is in the aviation profession) that the Russians didn't have the high-temp alloys like in the western world. This kept them down to lower temperatures, and thus their relatively fuel inefficient engines, at least in the past.
If anyone is particularly interested in the calculations of efficiency, look up the Brayton cycle in a thermodynamics textbook. I believe that most thermodynamics books will essentially evaluate the numbers are for the core, and don't give thrust numbers, rather they examine the shaft output power, but it's interesting to see, anyways, and it give an idea of the power output that goes to the bypass fan. BTW, the thermo book I have is "Thermodynamics: An Engineering Approach," by Cengel and Boles. You should be able to find it, or a similar book in a university library somewhere, if you're interested.
Also, to comment on another thread on this tech-ops forum, I wanted to say that although I don't frequent this forum quite as much (due to lower activity here), and although I haven't bothered to fill in my list of respected users (maybe I'll bother to do it later), I have to say I have great respect for the people here, and in the main forum, who are seriously interested in discussing information in a good, constructive manner. Thanks.