Basically what events led to the luftwaffe to becoming overpowered and stop bombing Britain.
The Battle of Britain was a protracted battle of attrition, a battle that Germany was not fully prepared for or geared towards but the UK had been anticipating in the 1930s. The Luftwaffe was extremely successful in the invasions of Poland and France, delivering concentrated surprise attacks that destroyed enemy air forces on the ground as well as in the air and, in conjunction with ground forces, dislocated their opponent's command and control systems to keep them off balance. As Stephen Bungay puts it in Most Dangerous Enemy, "Such blows are designed to create air superiority for a limited but crucial period of time. The ultimate expression of it is a tank on the runway."
Against the UK the Luftwaffe would have to operate independently, against a prepared opponent. The RAF had developed an effective command and control system that has come to be known as the Dowding System; radar is sometimes portrayed as a crucial British "secret weapon", and was a vital part of that system, but only a part.
During the First World War, when German Zeppelins and Gotha bombers started to mount air raids on Britain, the difficulties of air defence became apparent; defending fighters had trouble even finding bombers, let alone engaging them. As technology improved and bombers became faster, the problem became worse; Stanley Baldwin gave a famous speech in 1932 featuring the line "the bomber will always get through"; his reasoning:
"Take any large town you like in this island or on the Continent within such reach. For the defence of that town and its suburbs, you have to split up the air into sectors for defence. Calculate that the bombing aeroplanes will be at least 20,000 feet high in the air, and perhaps higher, and it is a matter of simple mathematical calculation—or I will omit the word "simple"—that you will have sectors of from 10 to hundreds of millions of cubic miles to defend. I beg pardon. I am not a mathematician, as the House will see. I mean tens or hundreds of cubic miles. Now imagine 100 cubic miles covered with cloud and fog, and you can calculate how many aeroplanes you would have to throw into that to have much chance of catching odd aeroplanes as they fly through it. It cannot be done, and there is no expert in Europe who will say that it can. "
Detection technology was mostly limited to observation; there was a fair amount of work with sound detectors such as [sound mirrors] (https://www.nationaltrust.org.uk/the-white-cliffs-of-dover/features/a-brief-history-of-sound-mirrors) on the south coast (photographs of Japanese mobile locators sometimes crop up as "[Japanese war tubas] (https://en.wikipedia.org/wiki/Acoustic_location#/media/File:Wartuba.jpg)"), but acoustic detection was imprecise and liable to be confused by anything from birds to somebody walking too heavily nearby. Without some form of early warning the RAF would have been forced to send up standing patrols of fighters hoping to bump into German bombers, placing massive strain on the fighter squadrons.
Inter-war experiments with radar, under Watson-Watt in the UK, offered a potential solution, and from 1937 the Chain Home network of RDF stations was established around the British coast. Radar was by no means an exclusively British technology; indeed the German Freya early warning radar in use at the start of the war was more sophisticated, operating at a higher frequency than Chain Home. As Watson-Watt said, though: "Give them the third best to go on with; the second best comes too late, the best never comes."
Chain Home allowed aircraft to be detected at long enough range to provide a useful warning, then, but that's only the first part of an air defence network. The right people need to know where your own aircraft are, where the enemy aircraft are, and be able to guide the former to the latter. It seems terribly obvious but it's the sort of thing easily neglected; not by Hugh Dowding, AOC RAF Fighter Command from its formation in 1936. Dowding established a network in which information flowed from Chain Home stations and an extensive network of Observer Corps posts (Chain Home had limitations in accurately determining raid size and altitude, and could not track aircraft over land after they crossed the coast) to a Filter Room. The Filter Room collected and assessed the data, then passed it on to an Operations Room. If you've seen pictures or film of WAAFs [moving wooden blocks around a map with long sticks] (https://en.wikipedia.org/wiki/Dowding_system#/media/File:Plotting_Table.jpg), that's an Ops Room. Fighter Command was divided geographically into Groups, each group comprising a number of Sectors ([map of Groups and Sectors, Spring 1941] (https://www.ibiblio.org/hyperwar/UN/UK/UK-Defence-UK/img/DefenseOfUK-45.jpg)). The Group assessed raids, and determined which squadrons to send up to meet them; these orders were passed to the Sector stations, who then directly controlled the squadrons by radio to vector them in on the enemy.
An attacking air force can pick and choose where it concentrates its attacks, giving it a tremendous advantage. The Dowding System was crucial to allow Fighter Command to get squadrons up at the right place and time to meet attacks, making most efficient use of its limited resources. It made it very difficult for the Luftwaffe to catch fighter squadrons on the ground. It was flexible, allowing squadrons to easily move between sectors and groups; Dowding rotated his squadrons to give pilots some respite, at least until attrition really began to bite in September. It wasn't perfect, but having been established before the war and tested and improved over the first months of 1940 it worked well during the Battle. As Adolf Galland put it after the war:
"From the first the British had an extraordinary advantage, never to be balanced out at any time during the whole war, which was their radar and fighter control network and organisation. It was for us a very bitter surprise. We had nothing like it. We could do no other than knock frontally against the outstandingly well organised and resolute direct defence of the British Isles."
(Galland rather overstates the case, perhaps deliberately; there was a German system that became increasingly sophisticated as Allied air attacks intensified, but the Luftwaffe were unaware of the extent of the Dowding System in 1940 and its importance.)
A second crucial factor was maintaining strength of aircraft and pilots. The total strength of the Luftwaffe committed to the Battle of Britain (some 3,600 aircraft) was greater than Fighter Command (around 1,000 Hurricanes and Spitfires), but the Luftwaffe total is aircraft of all types. Without being able to effectively bomb aircraft on the ground they had to be destroyed in the air, the job of fighters, and with around 1,000 single-engine Bf 109s the two forces were far more closely matched at the start of the battle. For replacements, though, Britain was producing twice the number of single engine fighters as Germany during the Battle, and had the great advantage of being able to recover pilots who bailed out or force landed. As the Battle progressed Fighter Command's strength remained broadly stable, while German fighter strength slightly fell. It was an area of great concern for Dowding, especially at the end of August and start of September, he was worried that losses were unsustainable, but the same was true (if not truer) for the Luftwaffe.
Compounding the issues for the Germans was faulty intelligence. The British overestimated the size of the Luftwaffe, but that was not the worst problem to have in a drawn-out battle of attrition, whereas the Germans underestimated the size of the RAF. Both sides greatly overestimated the number of kills scored, and that reinforced the German idea that Fighter Command was all but wiped out by September. According to Richard Overy's The Battle of Britain: Myth and Reality Goering was informed in early September that Fighter Command had been reduced to a strength of 100 serviceable fighters; on 6th September they actually had 738 operational aircraft, with a further 256 in stores ready for despatch.
(contd)