All those blindly saying "We travelled to Europe before the EU, we can carry on again" are completely ignoring the fact that the framework in which we fly has completely changed since then. Everyone else is remaining under the current framework, we are the only ones planning to leave it.
The aviation framework between countries was completely different and less free prior to the Open Skies agreement. Aviation regulation and the management of routes ("flying rights") was done bi-laterally between different nations. The Open Skies agreement created the right for EU carriers to access any routes in the EU, as well as the right for cross-border mergers of airline companies. The EU has also negotiated an Open Skies agreement with the USA/Canada, so transatlantic travel will also be affected. Negotiations over flying rights to the US routes have already begun with the US offering the UK a less favourable arrangement than that which we currently enjoy as EU members.
If the UK does not remain signed up to the Open Skies agreement, it falls back on previous bi-lateral agreements where they still exist. These are more limited in the rights they give to domestic carriers to access flying routes. For example, many of the smaller airports that airlines such as Ryanair use were not subject to these earlier agreements.
The situation is also complicated by the fact that previous bi-lateral agreements were also largely negotiated on the basis of domestic carriers. Many of these carriers have now merged under EU rules, eg British Airways/Iberia/Aer Lingus are now all part of the International Airlines Group.
For anyone interested in the technicalities, some of the legal firms specialising in aviation have done briefing papers, eg here
Who knows what is going on behind the scenes, but I think the airlines and travel companies are wise to protect themselves against any eventuality. Burying your head in the sand is a risky strategy.