The GDP (Gross domestic product) is not how much wealth a country has, but how much a country produces or earns in one year. So the colonies of the UK and France do not play a role, because they lost them decades ago (sometimes after long and bloody wars) and therefore they do not directly contribute this year.
The GPD can be disaggregated into two figures: GDP per capita and population.
In terms of GDP per capita, Germany is NOT particularly wealthy. Luxembourg, Liechtenstein, Norway, Switzerland, Ireland, the Netherlands, Sweden and Austria have a higher GDP per capita on a purchasing power parity basis (https://www.cia.gov/library/publications/the-world-factbook/rankorder/2004rank.html) than Germany, and its GDP per capita is also lower than the one of the USA for example. Germany does have a higher GDP per capita than France and the UK, however. It is noteworthy that Germany has such a slightly above-peers GDP per capita despite WWII, but Germany’s GDP per capita is not magical or anything and has been roughly on par with its large western neighbors for most of its history.
Population, the other important factor, does the trick. Germany has roughly 80 million citizens. Luxembourg, Liechtenstein, Norway, Switzerland, Ireland, the Netherlands, Sweden and Austria are far smaller. Even the UK and France cannot compete (https://www.cia.gov/library/publications/resources/the-world-factbook/rankorder/2119rank.html). So the reason why Germany is considered "rich" by some is not because its citizens earn a great lot of money have a much higher than average income, but because there are so many of them. Personally I would not consider that "rich", but it is true that Germany is a very strong economy.
Tl;dr: Germany is not particularly rich, it’s big.
EDIT: Clarification based on SkloTheNoob's comment and some small syntax corrections.
/u/tinrond has already made the important distinction between GDP and GDP/capita, the latter of which, as he explained, is not crazily above that of Italy, France or the UK. It should be mentioned, however, that Germany's GDP/capita today is a lot lower that it would have been had reunification never taken place. Indeed the Western states (especially Hamburg, Baden Württemberg and Bavaria) are significantly wealthier than their Eastern counterparts and in fact often much wealthier than comparable regions in other Western European countries.
However there are a couple of reasons that economists and historians often cite for Germany having such a robust economy compared to a lot of those other countries, some of which I'll briefly list here. I'm mainly concentrating on West Germany, because I know far less about the East. Being neither a professional historian, nor an economist, however, I'm quite happy if someone feels I've missed any out or have included a factor that may not entirely work.
Harmonious industrial relations - compared to the strikes that plagued Britain in the 1970s and France even today, Germany has had since the late 1950s very few strikes or incidents of severe industrial action. This is partly due to the Godesberg Program of the 1959 in which the foremost left-wing party in Western Germany (the SPD) dropped its commitment to replacing capitalism with democratic socialism, and instead sought to work with the capitalist system, making sure that trade unions had representation on the boards of companies. This had the end result of firms and trade unions collaborated and compromised with one another as much as possible, neither taking liberties, and making halts to production less likely.
The Mittelstand - Despite the existence of large, multinational firms originating from Germany like Siemens and Volkswagen people often maintain that the heart of the German economy lies in the so-called Mittelstand. These are small-firms, family-owned, often in small and medium-sized towns producing often high-quality specialist equipment. Because of this specialisation and high-quality of production German firms often have a very good reputation around the world in their particular field. Similarly, in contrast to the more bombastic world of Anglo-American capitalism, Mittelstand firms are far more conservative with their money, being family-owned after all, and less-willing to make extravagant investments or buy-out other companies, meaning more is often spent on R&D, leading to further innovation.
The school system - Okay, this is a controversial one. I'm not saying I support the German education system, however I do feel that it has contributed to the economic success of West Germany. It also depends on which state, as different states have often quite radically different systems, so I might be generalising a bit. However on the whole Germany's education system is tiered, so that depending on your academic success you can either go to a Gymnasium, Realschule or Hauptschule. I'm grossly simplifying, but generally Gymnasien are for the more academic and for those deemed university-bound, Realschulen for future technicians and Hauptschulen which are more vocational. This combined with an excellent system of apprenticeships means that most Germans are specialised to a particular or job when/if they come out of school at 18 and are able very quickly to get a job.