There's actually a rather short convincing answer; with a few blips and exceptions, success in european football competitions follows a fairly simple pattern, and can be divided into four post-war eras. They overlap slightly, but I've given them precise start and end dates for convenience sake:

Period 1: 1955 to 1962 "The Dictator Era"

Real Madrid Club de Fútbol was Spanish Dictator Francisco Franco's poster-child for the success and legitimacy of his regime. Because how bad can a guy really be if his country's got a successful football club, right? (/s) Franco funneled resources into "Real," creating a vital tool through which the Spanish could feel they were connected with (and better than) the rest of Europe. Sure, the disgruntled working classes of Madrid preferred supporting Real's more humble crosstown rivals, Club Atletico de Madrid, but the middle classes were arguably even better poised to appreciate the prestige and importance of international success. It's also important to note that the other two European Cups won in this period went to Sport Lisboa e Benfica, a team which played a similar role for the Portuguese dictator António de Oliveira Salazar.

Period 2: 1962 to 1992 "The Iron and Industry Era"

It would be unwise to write off the wonders that a supportive dictator continued to do for a football club, seeing as various dictatorships had their favorite clubs appear in eight more finals; basically one every other year until the "Third Wave" of democratization began in the mid-1970s (with contenders from Spain, Portugal, Jugoslavia, and Greece during the rule of the "Colonels"). The 1966 final between Partizan Belgrade and Real Madrid stands out as having made not one but two dictatorial regimes very happy.

But despite strong showings, dictators would only be able to taste success three times in these thirty years (1966, 1985, and 1992) because new force in european football had emerged. The prosperity of the post-war Europe saw large population migrations in urban centers, increased standards of living, all a manner of products marketed to consumers, and the emergence of mass media: from factory workers in Milan and Manchester to dockworkers in Amsterdam and Liverpool, all over Europe people ended the workweek thirsty for beer and football. And they often ran out of beer halfway through Sunday, leaving the stadium the only viable option to keep their buzz going. (/s) Advertisers rushed to football clubs and their packed stadiums, and clubs (and players... and agents) charged appearance fees to release their star players for photoshoots and TV appearances.

This era was dominated by clubs from industrial centers: Milan, Turin, Munich, Amsterdam, Manchester, Liverpool, and even Glasgow sent teams to European finals. Heck, even as Spain began to liberalize its economy in the 1970s, Madrid's working-class club Atletico de Madrid even appeared in a final. France sent a club to a final for the first time: Saint-Étienne, a club France's most important mining center. In Germany, in addition to the Bayern München dynasty in the mid-1970s even the smaller club Borussia Mönchengladbach near the industrial Ruhr region was able to claw its way to a final. When a Portuguese club again won a final after 25 years, the honor went to the team from the industrial shipping center Porto, and not previous winner Benfica.

Period 3: 1992 to 2007 "Soccer Goes Global and the Information Age"

Satellite communication, the fall of the Berlin Wall, "Made in China," cable TV, the Internet. Postmodernity was invented in the 90s, and not everyone was okay with it.

Aston Villa, Nottingham Forest, Hamburg, Leeds; these teams would be seemingly forever relegated to mediocrity as manufacturing evaporated and/or advertisers and sponsors moved elsewhere. But while some fell behind, new money seemed to be floating down from the sky for others. More succulently, some some cities adapted, others didn't. The city of Milan transitioned from being Italy's industrial center to its media, finance, and service hub, connected to the rest of Europe by highways, railways, and two airports; unsurprisingly its first team was gobbled up by a media empire its second by an oil tycoon. New money pouring in from advertisers carried AC Milan to eight finals between 1988 and 2008 (one every two-three years, more or less). New money was flooding into Spain too, allowing clubs like Barcelona and Valencia that had been neglected by Franco's regime reach new heights as well as fueling Real Madrid's resurgence. Interconnectivity and global branding was the key to sustained success, sometimes in surprising ways: the fact that the cities Marseilles and Genoa are separated by one of the most-visited beachfront regions in the world definetely contributed to the flow of business that sustained Olympique de Marseille and U.C. Sampdoria, allowing them to appear in the 1991 and 1992 finals respectively (and Marseille would win after appearing again in 1993). Interconnectivity was also an important factor allowing clubs to make use of innovative sports research facilities and attracting competent staff both in the locker room and the boardroom.

Period 4: 2007 to today "Banker Wankers and Management Consultants"

What do Paris, Rome, Madrid, and London have in common? They're all capitals. And capitals attract capital. As teams with a combination of globally marketable brands and shrewd business sense had come to dominate football in the previous era, now a new wave of investment was flooding into football that took a more holistic approach to investment. Investors ranged from Arab princes to Russian oligarchs all the way to American entertainment groups. Some wanted to invest in brands they believed they could sell (like the American investors who bought Manchester United and Liverpool) while others wanted brands they thought they could build up (like the investors who bought AS Roma, which had only ever appeared in one fluke final in the 1980s). Others wanted a base to conduct more business in the country (like the owners of Chelsea FC in London and Paris Saint-Germain in France). As football becomes a bigger business, people are only going to invest clubs already located in respectable business hub (incidentally, this is why now you see such a strong shift in resources to Southern English clubs in the English Premier League). Of course, there are a few exceptions: Munich certainly isn't the only business hub in Germany, but they did a good job early on developing a strong brand (and their brand certainly got a boost early on thanks to a stronger "Tribalist" sentiment in Bavaria) and a sensible management structure made them real trendsetters in Germany. Barcelona has a similar story. AS Roma is a more peculiar case, but evidently the new management came in hoping to exploit the city's renowned brand to propel the club into global notoriety.

In any case, if you'd like to read more the greatest book examining the "Economic History" of football is Soccernomics, by Simon Kuper and Stefan Szymanski, and where I got nearly everything written above from.