TL;DR version: You can probably accept that advice, although the extent of the effect is probably smaller than TLC is making it out to be.
First, a little "theory" just to set some ground rules.
Refrigerators effectively remove heat from items, and expel that heat (and additional heat generated in the process) out the "back". They do this by cycling "cold" air through the compartment, allowing that air to heat up (very slightly) from the contents (if the contents are warmer than the chilled air), and then cycling that warmed air across cooling coils. See http://home.howstuffworks.com/refrigerator2.htm for example. They regulate this process to keep the chilled-air temperature relatively constant - within a few degrees. Similar to your air conditioner, if the chilled-air temperature is above the allowed range, the compressor and thus cooling system will "kick on", and will stay on until the temperature reaches the target temperature at the bottom of the allowed range.
Air is a rather good heat transfer medium. The reasons for this are that (1) heated air can be easily physically moved to the cooling system, (2) turbulent flow is relatively efficient to achieve, and (3) materials tend to have high heat transfer coefficients with air. It's also (fortunately), relatively unlikely to contaminate the materials being cooled.
Defining the Question
Now, the question on "efficiency" hinges on what we expect our refrigerators to accomplish. If what we want is to just put stuff in it, wait some period of time, and efficiently keep that "stuff" cold, we will arrive at a different answer than if what we want is to efficiently make some "stuff" cold, regardless of how efficient it is to keep it cold. The reason is that in the former case the heat transfer we are worried about is between the chilled air and the insulated refrigerator housing, while in the latter case the heat transfer which matters is between the warm food items and the chilled air. A third case, alluded to in your question, is how well we expect the former case (keeping chilled foods cold) to behave when one of the six walls of the container is completely removed (the door is opened).
Scenario: Chilling a Warm Dish of Food
I'll hit the middle case first. Placing a warm dish into an empty refrigerator (filled only with circulating air) will chill that dish far more rapidly than placing the same warm dish into a block of "chilled" dishes in the fridge, because the heat transfer coefficient of warm dish to chilled dish is much lower (less heat will flow out of the warm dish) than the heat transfer coefficient of warm dish to turbulent air flow. However, this is a hard difference to observe without really trying at it, because even a relatively "full" refrigerator will have copious air flow around any inserted item; it applies more to a freezer where you might be able to pack a bunch of regularly-shaped boxes tightly together. So, in this case, the answer is that an empty fridge is more efficient at cooling, but that verdict is not typically relevant. Also, it doesn't figure into typical home refrigerator "efficiency" as related to your electric bill because the temperature of the chilled air is typically kept constant rather than acting in a blast-chiller mode (where the air is chilled far below the target temperature of the food for a short period of time to bring the food quickly down to temperature).
Scenario: Door Shut, Foods Chilled; Keeping It Cool
When the system is in stasis - the solids inside have all been chilled - the effects of "empty space" are debatable. In a simple refrigerator with fixed-flow air circulation fans, circulating air through mostly-filled space will result in greater turbulence and thus higher heat transfer coefficients and higher potential loss of heat at the edges. However, the operational heat resistance of the refrigerator's shell should be primarily due to its insulation - the overriding barrier to heat transfer is not the transfer from the inner surface to the chilled air, but the transfer through the insulated foam layers between the "outside" and the "inside" surfaces. The exception here is where there are insulative weak points - for instance, poorly-fitting gaskets around doors and inlets which stop mass air flow but have much less resistance to heat transfer than the full gasket. Overall, though, this will be very minor, and unlikely to cause any measurable difference in your electric bill. At stasis, then, there is generally no difference in efficiency.
Scenario: Where Is the Mustard?
Last case is when the door opens. This is the main cause of lost "chill" / heat inflow in any real-world refrigerator. This is no longer a simple "heat transfer" problem, but a mass transfer heat exchange: chilled air flows out, and warm exterior air rushes in to replace it. Home refrigerator designs aim to minimize this transfer, even at the expense of their efficiency in the first case above (blast-chilling foods). They do this by keeping the circulating air currents parallel to the door so that an open door does not directly expel chilled air. They also limit the velocity of air circulation to keep it "mostly" still when the door is opened.
Assuming absolutely still air, though, there is still a significant amount of mass-based heat loss through the open door: cooler air is denser and will thus tend to flow out the (vertical surface) opening, allowing warmer exterior air to flow in from above. The impact of this depends on how long you keep the door open: eventually the cooling system will reach a new equilibrium, chilling air and sending it out the bottom portion of the door, while the inside of the fridge above the air circulator will become essentially room temperature (which itself is warmed from the "other half" of the coolant system in the back of the fridge).
So: does a "more full" fridge affect this mass air flow? Yes, it could. In the "still air" extreme, a "full" fridge will contain less air, and so less air will flow out, even after a few minutes of the door being opened. Unless you tip your fridge so everything spills out, the "cold sinks" of the chilled foods will remain inside even if all the chilled air around them evacuates; when you close the door again the coolant system has less work to do to bring the now-room-temperature internal air back down to chilled temperatures because there is just less air to chill.
In the "forced circulation" extreme, though, there is a distinct possibility that your food items are altering the designed air flow in the fridge, causing an outward flow of the chilled air rather than a door-parallel "curtain" flow. This isn't necessarily a factor of how much food is in there, but on the specific placement and shapes involved. Another factor is that if the fridge is virtually packed, the "chilled" air will be moving faster (assuming a constant-volume circulating fan, which is to be expected), and thus any leakage along even the door-parallel curtain flow will be increased.
We end up with a little bit of a mixed bag there. Having a more "filled" fridge will cause more chilled air to leave the fridge while the door is opened; however, having a more "filled" fridge will allow the replacement room temperature air to chill to refrigerator temperatures more quickly. My guess is that in "normal" circumstances it is probably a safe bet to say that the latter outweighs the former in terms of efficiency, and the advice is "correct". However, to get a real measure of how important it is for your fridge you'd need to run an experiment with a electricity meter.
My guess is that the effect of the fridge being filled is probably not the most important aspect of that appliance's electricity bill impact, but that it is also probably significant. If you want to take TLC's advice, fill the "empty spaces" in the fridge or freezer with low heat transfer coefficients which will thus retain "coolness" longer when the door is left open. But, any effect is secondary to the primary cause of losing your cool: leaving the door open in the first place. Get in; get out. Know what is in there, and where it is. If having a bunch of empty tupperware or campfire rocks in there gets in your way and makes your fridge-gazing take longer, then it's completely backfired. I think the more practical advice is simply: keep the fridge organized, and know when you are out of something. Then, you won't spend several minutes trying to find something that isn't there.