Why recover with carbohydrates, instead of with protein?
This answer is primarily based on 2 books by Chris Carmichael. The Time Crunched Cyclist and Food for Fitness, plus some other sources that I am unable to dig up at the moment. 

Here's the gist of it... Following a race, intense workout, or an endurance ride, the muscles and liver are glycogen depleted. The key to recovery is to replenish glycogen so that you begin your next event with full glycogen stores. The major building block of glycogen is carbohydrate. The body quickly and easily disassembles carbohydrates into glucose and that glucose is readily available for conversion into glycogen by both the liver and muscle cells. Protein is synthesized into glycogen, however protein must be first transported to the liver, converted to glucose, and then transported to the muscles. So, restoring glycogen levels with protein is less efficient and more prolonged than restoring glycogen with carbohydrates. 

According to Carmichael, the 3 critical components post-ride are carbohydrate, sodium/electrolytes, and fluid. And, your body is primed to replenish muscle glycogen stores most rapidly within the first 30 to 60 minutes following exercise, and the sooner the better. As GuyZee points out in another answer, "the glycogen window". 

Back to protein... Protein is also important for recovery, primarily for muscle tissue repair and additionally protein enhances the ability of muscle tissue to utilize carbohydrate in the replenishment of glycogen. A general rule of thumb is a 3:1 carb to protein ratio; but some experts recommend a 4:1 ratio. 

In essence:
Carbs for glycogen replenishment.
Protein for muscle repair.
Electrolytes and fluid.  

Examples... One would be something like a bowl of rice and sauteed vegetables plus chicken breasts. And for a "technical" product, something like Recoverite by Hammer Nutrition. I personally like the Hammer products as they don't use sucrose and high fructose corn syrup; instead a primary ingredient is maltodextrin which is basically long chains of glucose. Sometimes I don't have time for an actual meal, so I'll use Recoverite in that case. It can also be chocolate milk. The basic idea is that you need to replace and repair.

FWIW - None of this matters for the casual cyclist or the "typical commuter". Just eat normal, healthy meals and you'll be fine. 

A few references:


REGULATION OF MUSCLE GLYCOGEN REPLETION, MUSCLE PROTEIN SYNTHESIS AND REPAIR FOLLOWING EXERCISE (International Society of Sports Nutrition Symposium)
RECOVERY - A CRUCIAL COMPONENT FOR ATHLETIC SUCCESS (Hammer nutrition)
How to Plan Post-Exercise Recovery Meals (livestrong.com)