Why are aeroplanes not made in one place? As an example, different parts of an Airbus A380 are produced in factories all over Europe, and then assembled in a factory in France. Transporting these massive objects must be incredibly expensive (unless in fact they are shipped in small parts).

Why not build it in one place, like ships?
Short answer
Airbus is a particular case of decentralised supply chain which came from historical and political reasons. Airbus is a consortium with several European states having a stake on it, which means that each state wants a part of the production in their country.
Long answer: Why is Airbus' supply chain disperse
The aviation industry for commercial planes basically has has two large players nowadays: Boeing and Airbus. The 70's-80's saw the consolidation of American producers, with Boeing as the survivor and most successful of them (incorporating other large players such as the old McDonell Douglas).
Europe had smaller producers, such as BAC or Aerospatiale. Smaller European producers in the 70's and 80's wanted to compete with Boeing, namely with the success of the 727, but the fragmented market would be a challenge. Therefore, Airbus was a consortium created with several European manufacturers to have a large aviation player with European roots. To minimize political difficulties, the supply chain was split over several countries:

While the original partners in the company recognised that the only way to compete was to consolidate national industries on a regional European basis, consolidation required negotiation and political finesse. Most sensitive was the decision where to site the different manufacturing operations.
The easiest way to defuse tensions was to build different bits of the first plane, the A300B, in the different partner countries. The French made the cockpit, the control systems and the lower-centre section of the fuselage; the UK made the wings, and the Germans made the rest of the fuselage and a part of the centre section. The Dutch made the moving parts of the wing, the flaps and the spoilers, while the Spanish made the horizontal tailplane.

Source: Financial Times
This supply chain is unlikely to become more centralised or placed offshore, for political and economic reasons. 

From the economic perspective, today's supply chain is too fixed to be able to be moved somewhere cheaper, as the regions where each part is produced developed local expertise and innovation centres around it. Another argument is that the de-centralized supply chain helps to have a faster delivery time (see FT article mentioned above).

From the political perspective, European stability (commercial and military) requires that countries depend on each other for developing aircraft. Although Airbus is mostly known from the A380s and A320s, they also produce military aircraft with their subdivision Airbus Defense and Space. Placing production outside of Europe would place aviation expertise in other countries, a strategic expertise to have for defense.