Who said Those who do not learn from the past are doomed to repeat it?

Edmund Burke (1729-1797) said "Those who don't know history are destined to repeat it."

George Santayana (1863-1952) said "Those who cannot remember the past are condemned to repeat it."

The full quote from The Life of Reason (1905-1906) by George Santayana is as follows:

"Progress, far from consisting in change, depends on retentiveness. When change is absolute there remains no being to improve and no direction is set for possible improvement: and when experience is not retained, as among savages, infancy is perpetual. Those who cannot remember the past are condemned to repeat it."

(Many other people have repeated this, or paraphrased it, and the original quote is often misattributed to Winston Churchill, but the quotation is unable to be found in his written work.)
"Winston Churchill " It was actually George Santayana. He was a Spanish philosopher.

Actually the author originally credited with any such phrase is Edmund Burke. He died before either Winston Churchill or George Santayana were even born.
Simplest Answer. Another way of wording the famous quote is: "learn from your mistakes."

The Long Answer:

As our species pulled it self out of the primordial ooze there has been a constant pattern to our growth. Throughout history, by evolution of the mind as well as technology, we have followed the same series of events leading to the same negative conclusion. Each time mankind has re-started the cycle the stakes of failure have increased.

To over simplify this series of events or cycle is as such: There exists several small communities, these communities join to create a large community, and this community begins to grow. Eventually it begins to consume large amounts of resources; natural and human, it grows to the point were it begins rising beyond it's means. This causes a need for more resources as they are consumed at an ever-increasing rate.

As history has shown us from the simplest village to complex social systems these needed resources are acquired by brutal acts of desperation and war. Eventually the system becomes ineffective and far too large to sustain its own growth and eventually it collapses upon itself creating casualties both human and environmental in this process.

Before our current cycle of human events the possible risk factor was rather minimal, but as we have grown into a global village the stakes are considerably higher. As we witnessed in the near conclusion of our current cycle at the end of World War 11 (a-bomb) and the event that followed (e.g. Cuban missile crisis, global warming, environmental degradation, etc.)

To simplify yet again as countries fell into the last stages of desperation war was inevitable. Rather then being between two countries or an act of civil war a conflict between several nations erupted, World War. This was made possible by this cycle's large stride in technology (planes, submarines, battleships etc.). Believing that the actions of the WW1 eventually led to WW11 we will treat them as a similar entity. Mass casualties strained the entire planet as every society and economy was affected.

Then in August of 1945 two Atomic Bombs (aka A-bomb, code named Big Boy and Fat Man) were dropped on the cities of Hiroshima and Nagasaki. Progress had eclipsed the human race as technological advancement surpassed human evolution. The stakes of failure then reached an unprecedented height, for the first time in the history of the world humans had the ability to erase all life on our planet.

Humanity was forced to re-evaluate its standings in the world as it began to realize all that had been lost and has done so several times over the last six decades. Civilization cannot afford to follow the cycle to its previous conclusion confirming, "Those who forget the past are doomed to repeat it". The two World Wars featured the most casualties (civilian/military) in history, bearing witness to these events, as a global voice, humanity declared "Never again".

"We stand on the shoulders of giants and owe a lot to them, however though we maybe only a few heads higher we can still see further. We have an obligation of Stewardship."

- Matthew Carter at Design Thinkers'09

"Our civilization, which subsumes most of its predecessors, is a great ship steaming at speed into the future. It travels faster, further, and more laden than any before. We may not be able to foresee every reef and hazard, but by reading her compass bearing and headway, by understanding her design, her safety record, and the abilities of her crew, we can, I think, plot a wise course between the narrows and the bergs looming ahead.

And I believe we must do this without delay, because there are too many shipwrecks behind us. The vessel we are now aboard is not merely the biggest of all time; it is also the only one left. The future of everything we have accomplished since our intelligence evolved will depend on the wisdom of our actions over the next few years. Like all creatures, humans have made their way in the world so far by trial and error; unlike other creatures, we have a presence so colossal that error is a luxury we can no longer afford. The world has grown too small to forgive us any big mistakes."

- A Short History of Progress, Ronal Wright
If we don't know the mistakes people in the past have made, we will make those same errors.