There's a few factors at play - not all of which were unique to the Germans - that led to these high kill counts.

The simplest factor - as well as the least controversial - is that these German aces were often kept at the front until killed, captured, or otherwise put out of action. Compare that will the practices of Allied Air Forces - particularly that of the United States. The US would regularly withdraw experienced pilots, especially those that achieved Ace status, with the intention of having them teach at flight schools. For the US, the decision was clear - they saw the value of experienced flight crews in teaching those in training and determined it to be more than keeping an expert pilot on the front. That's not to say that the decision of Axis powers to keep their aces on the front was so short-sighted - there were often manpower shortages in play that meant that they couldn't afford to withdraw these pilots. The Luftwaffe had infamously thrown away many of their instructors on the Stalingrad encirclement, but the nations with the next two highest-scoring aces had similar manpower problems. Japan's flight schools had an alarmingly small output for much of their existence, and this chokepoint and the manpower shortages it caused with flight crews was apparent as early as the 1939 Battle of Khalkhin Gol. As for the Finns - responsible for the highest-scoring non-German ace - I'm not familiar with the specifics of their pilot training, but the Finns across the board were suffering from critical shortages in many key areas during WW2.

Following along this same trend, the Germans were operating in a target-rich environment with very few aircraft of their own. The general trend of the war was that German aircraft were increasingly outnumbered, both due to higher production on the side of the Allies and fuel shortages on their end. Non-fighter aircraft were increasingly vulnerable as Allied fighters became more numerous, so the Luftwaffe would withdraw them and replace them with Fw 190s working as fighter-bombers. And while the Luftwaffe began drawing down its aircraft most vulnerable to fighters, the opposite trend occurred with the Allies. The British and Americans regularly sent hundreds of bombers over German cities. The Soviets fielded air armies capable of putting several hundred aircraft over a stretch of front. This was particularly relevant, as all of the highest-scoring German aces achieved this on the Eastern Front. While the Soviets were reaching a point where they could keep a continuous presence over the battlefield, the increasingly beleaguered Luftwaffe was forced to relying on fire brigades, where air assets were pooled and rushed to crisis points on the front. That meant that the top pilots were pretty much always fighting, and fighting intensely. For reference, in the 1944 Crimea campaign, Gerhard Barkhorn's II./JG 52 had pilots flying as many as five sorties a day against Soviet aircraft. This kind of density of targets is also why we see the incredible claims of Finnish pilots - the sparse Finnish Air Force was operating against huge numbers of Soviet aircraft.

And that leads us to one of the more controversial reasons behind these scores. In all likelihood, many of these top scores are exaggerated. This is not something exclusive to the Luftwaffe, nor does it signal any malicious intent on the part of the pilots. Over-claiming victories is common among pilots, and the general trend tends to be that over-claiming kills increase as pilots are overworked or otherwise stressed. For an Allied example, look to the Flying Tigers, which claimed more victories than the Japanese ever had aircraft in the theater. For the Germans, they generally had more realistic counts early in the war, as they were often overrunning crash sites. But late in the war, kill claims often became inflated to an absurd degree. In the offensive phase of Kursk, German claims were almost double the reported Soviet losses. During Barkhorn's time in the Crimea in 1944, for example, the Luftwaffe claimed more than the 1,200 aircraft the Soviets had committed to the operation, while the Soviets listed only 179 aircraft lost.

All that being said, the clear incidents of over-claiming victories doesn't mean that the Luftwaffe wasn't often superior to its foes. Especially on the Eastern Front, Luftwaffe fighter pilots were often significantly better trained and equipped than their adversaries in the Soviet Air Force. Up until 1942, for example, the Soviets never really had a fighter that could compete on even terms with the Bf 109. Soviet aircraft often lacked radios, which hindered formation flying that is crucial to aerial warfare. Meanwhile, German pilots were trained well both on an individual level and in group tactics. Obviously, these trends became less relevant as Soviet equipment and training improved and attrition took its toll on Luftwaffe crews, but the quality disparity was a very real factor, at least on the Eastern Front where the bulk of these claims by top Luftwaffe aces were made.