The Answer May Be Because Your Brain Is Smaller!
Dr. Chiara Cirelli and her colleague Dr. Giulio Tononi both of the University of Wisconsin Center for Sleep and Consciousness introduced a theory in 2003 that during sleep the brain was restored through a process called Synaptic Homeostasis (SHY).
Their hypothesis was that during sleep the brain takes advantage of decreased sensory traffic and uses the time to sample all of the synapses (the connections between neurons) and "renormalize" them in a smart way. They further suggested that without sleep synapses become overloaded as the day goes on causing them to "burn out" much like an electrical circuit with too many devices plugged into it.
Now, after nearly 14 years of study, Dr.'s Cirelli and Tononi have published the results of a study they performed on mice which seems to prove their theory. They have reported that during sleep this restorative SHY process actually showed the brain's synapses to shrink by nearly 20 percent! The results were verified using an electron microscope. So it seems that the restorative process in which the brain actually shrinks, resets the neurons back to a state that results in increased cognitive function when we awake after rest.
Dr. Russell Foster of the Sleep and Circadian Neuroscience Institute at the University of Oxford has reviewed the study and call's it a "very nice, clear piece of work".
To be sure there are many other reasons why we benefit from sleep but it seems that SHY is a big factor. Is your brain getting too big? Get some sleep and shrink it! You may feel a lot smarter as a result!