There were a number of forces, people, and events that led to the Allies having many heavy bombers while the Axis had relatively few. Also, I will be focusing on the US, UK, and Germany, as I am most familiar with those nations.
People
The first was a bit of happenstance. While the main theorist of strategic bombing was the Italian Giulio Douhet, he was joined in his advocacy for the practice by other influential people in other countries. The US featured Billy Mitchell, the UK had Hugh Trenchard, and Walther Wever led the German contingent. Wever died in 1936, which left Germany without its biggest advocate for strategic bombing. Trenchard and Mitchell continued their careers as officers up to and through WWII--always pushing for the bomber force. Having such strong personalities while the Germans were lacking certainly contributed to strategic bombing being more central to the plans of the UK and US.
Forces
The main forces involved in the disparity between the two forces were the differences in relative industrial capacity. In the below quote, Martin Van Creveld analyzes aircraft production for the five principal combatants.
Taking the period 1940-45 as the best available compromise, we find that, in those six years, the United States produced 309,761 aircraft. Germany built 109,586, the British Empire 143,234, the Soviet Union 147,836, and Japan 74,646. Earlier in this chapter it was estimated that, if America's industrial potential in 1939 stood at 3, then the figures for Germany, the British Empire, the Soviet Union, and Japan were 1.2, 1, 0.8, and 0.5 respectively. Combining the two sets we find that Germany produced somewhat less than its fair share of aircraft. The British Empire produced considerably more, and the USSR many more. Japan, too, produced more than its share. This calculation ignores many factors, including the extent to which the various countries mobilized their resources and the types of aircraft each produced. Still we conclude that, relative to their overall industrial potential, all the main belligerents except Germany built more aircraft than the United States did.
So, you can see that the US and UK had a sizable advantage on German production. This includes the fact that the US and UK produced more complicated machines such as four engined heavy bombers. With the UK producing roughly 1.5 aircraft for every German one and the Americans producing at a roughly 3:1 ratio, you can see that they could afford to spend relatively lavishly on their bomber force. The geographical barriers of the English Channel and the Atlantic ocean also made the idea of long-range bombers attractive to the UK and US respectively. Russia was pushed to the brink on the ground by Germany, and had not built up their air force prior to WWII to the extent that it could employ strategic bombers in any great numbers. (I must admit that I am less well read on Soviet doctrine at this time) You can see from Japan's production that it could not afford the luxury of expensive four engine heavy bombers. They did, however, get great results from their long-ranged two engine bombers. Italian production never matched Douhet's desires for his nation.
Events
One wouldn't normally think to look to the Spanish Civil War for heavy bombers, especially since the huge four engined beasts of WWII were absent from the conflict. However, the different combatants of WWII each observed the war in one way or another, and each took away their own lessons from the conflict. The Germans employed the Condor Legion in Spain, and while they had a good deal of success in employing level bombers against strategic targets, they also saw that their bombers were not invulnerable. Despite only relatively light fighter opposition, German bombers were not able to always get to their targets. As such, German aerial theorists shifted to emphasize tactical air power over strategic uses. The British and Americans saw the same air battles, but deduced that the two engined German bombers were insufficiently armed and not robust enough to do the job--but the big four engine bombers would surely be able to accomplish their missions. As WWII would illustrate, bombers could not survive unescorted. The British learned this fairly early on in the conflict, while the US continued to think they could do so until 1943.
Van Creveld also points out that the US and UK had a lot of what we could call "sunk costs" into heavy bombers by the time hostilities began. Influential military officers and politicians continued to push for the heavy bomber--especially in the US--throughout the war. The argument advanced by these advocates was that:
These bombing advocates dismissed the idea that the victims of such attacks would be more determined to resist rather than have their morale shattered. In reality, targeted civilians were remarkably resilient to the supposed shocks to their morale.
There were prewar advocates for heavy strategic bombers in a number of nations before WWII began. Only in the US and UK were such fleets constructed, which was facilitated by their prodigious industrial capacities. Germany focused their resources on tactical bombing instead. These three nations took very different lessons from the experiments in strategic bombing during the Spanish Civil War as well.