The main reason is that the European Union (and before that the European Communities) was intended by its architects to bring about ever tighter integration, possibly all the way to a federal state. Seemingly technical measures like the customs union and common market were not only or mainly intended at boosting economic output but at fostering understanding between people “from below”.
This integration was always meant to be a continuous and ongoing process. While it didn't have a common currency, the 2001 European Union was already very different from the 1957 European Economic Community, with the merger of the three European Communities, the directly-elected European Parliament, (mostly failed) attempts at creating a common defense or foreign policy, the police and justice cooperation, the Schengen border-free area, the “EU citizenship” and broader free-movement rights established by the Maastricht treaty, etc.
A single currency would therefore appear like the logical next step, a huge symbol of integration and cooperation. It's possible to find some weak economical arguments for it (enforcing fiscal discipline, preventing currency fluctuations, etc.) but I don't think they were decisive in the process leading to the creation of the euro.
Jacques Delors, who was actually very enthusiastic about the process, famously said that “Europe is like a bicycle. It has to move forward. If it stops, it will fall over.” So the Euro is perhaps the most visible and the most consequential – in a negative sense – of these but it's part of a series of initiatives to deepen the EU that were widely seen as necessary, not in a narrow technical sense, but for the sake of the European project itself.