Armchair physicists are quick to point out that this is nonsense, because a steel frame is a nearly ideal spring, and it will return any energy you put into it. They will point out that you can put a frame on a work bench, and flex it 10,000 times, and have almost no heat buildup in the frame. This "proves" that there's no energy loss from pedaling on a flexible frame.
Armchair physicists tend to miss the bigger picture. (Except for me of course).
What they got wrong is that the energy loss isn't absorbed by the frame. Instead it is absorbed by your leg muscles. While a frame is an ideal spring, your legs aren't. If you push with your legs, you use energy. If something else pushes back on your legs, you still lose energy. This happens because of the way your muscles work internally. The short answer is that there is no way for your muscles to turn mechanical energy into glucose (or any chemical that provides fuel to your muscles).
So don't all frames flex some? And don't you push with the same force regardless of how much the frame flexes? Yes. But here you need to understand some basic physics. Work is defined in physics as force × distance. If you push on something (force), and it moves (distance), then you've done work. Intuitively, work is the consumption of energy. Conversely, energy (as defined in physics) is the ability to do work. The units of work and energy are identical, because in effect they measure the same thing from different ends.
But wait a minute, if you push on a wall, and it doesn't move, you're still doing work. You get tired don't you? You do indeed. The wall doesn't move, however your muscles do. In fact each individual muscle fiber is constantly in motion, flexing and relaxing. The amount they do this when your muscles are relaxed is your muscle tone. Better muscle tone consumes more energy just sitting on a couch. When you actually use your muscles, they twitch farther and faster, and more work is done (internally to the muscle) and more energy is consumed.
At any rate, If you push on a spring, instead of a wall, and then relax your muscle, the spring will push your arm back to where it started. The spring will release nearly all of its energy back into your arm. However, your arm can't do a damn thing with it. All it does is make your arm warmer. That's where the energy loss takes place. This brings me back to the question of "don't all bike frames flex?". Yes, but if you push with 10 pounds of force on a hard spring, and it moves 1 inch, you've done 10 inch-pounds of work -- loaded the spring with that much energy. When you release the spring, that energy turns into heat in your arm and can't be used again. If you push on a soft spring and it moves 10 inches, then you've done 100 inch-pounds of work, and that much energy is lost. So a softer spring wastes more energy than a hard spring.
So how do you know that when the bike frame "unflexes" it doesn't send that energy to the road, rather than your legs? Well, that would be a tough problem to solve. But it stands to reason that some of the energy pushes your muscle, and some of it pushes the road. So you probably don't lose 100% of the energy you loaded into the frame, but you almost certainly lose some of it.
[O.K. I admit that the spring calculations are wrong. You can't push a spring through ten inches with 10 pounds of force. If ten pounds is the force used when fully compressed, then you push with five pounds when it is halfway compressed, and zero pounds when it isn't compressed at all. So you really want to integrate across the compression stroke, but since springs compress nearly linearly, this works out to a simple average, in other words for a final force of 10 pounds, the average force through the stroke was 5 pounds. The conclusion you reach, that softer springs waste more energy, is the same.]
On the other hand, there is still a reason why you may want to avoid stiff frames.
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