The process is called "annealing". I have worked metals for many years, used to teach jewelry making, silversmithing, pewtersmithing, etc for over 15 years.
When you hammer, or even bend most metals they do harden...it's called "work hardening". Pewter is an exception, it actually work softens...but that's another lesson.
I agree that the heated copper should be quenched in water, wait until the red goes away as some metals will crack if quenched when too hot although I have never experienced that with copper; brass, sterling silver...oh yeah, they'll crack. As they say, "don't ask me how I know."
Also, copper isn't the easiest metal to melt. If you are only using one of those Bernz-O-Matic type (hand held propane torches), don't worry you won't melt the copper. I do agree that looking for the red in a darkened area of the garage is a high enough heat to anneal copper. It won't harm the copper if you take it to a higher heat but it may cause some fire scale to form on the surface. Fire scale will be a dark area of the copper and it is actually harder than the copper below it. Fire scale comes off copper easily. Jewelers use a solution we refer to as "pickle". Pickeling solution used to be a dilute sulfuric acid solution back in the day but most jewelers today use sodium bisulphate ("PH Down" if you are a swimming pool person). With copper and what you are doing, you could use cold pickle. A hot solution removes fire scale more quickly but like I said, fire scale comes off of copper rather easily. If you do get fire scale regular old white vinegar should remove it quite handily. Though I've never really used it but vinegar is mildly acidic.
I must disagree that it will return it to it's original thickness, it simply softens the copper and allows it to be compressed again on the next use...They can only be reused so many times before they become too thin to be of any use.
OK, teaching mode off......