Over the 40 plus years in which the UK has been involved in the Europe project, our industries have become increasing European / internationally orientated, by which I mean working across internal European borders / and owned across European / international borders.
The existing Open Skies agreement is between the USA and the European Union / EEA, and our principal transatlantic international airlines are European owned, so under the agreement can operate between UK and the USA, even though the majority shareholders may not be British.
The USA is going to insist that the airlines involved in any new USA / UK Open Skies agreement must be majority USA or UK owned - easy for the USA, which I believe bans majority foreign ownership of the carriers, much harder / impossible for the UK, and in Trump's view of world "free" trade that's fine by him.
Few people who voted in the referendum would have given a second thought to trivial maters like international air agreements; but the thought of possibly fewer flights, less competition and higher fares may make them wonder what they voted for. Once wonders how the likes of our two biggest tour operators and their associated airlines, neither UK owned, will cope trying to offer holidays to Florida, for example from the host of UK airports from which they currently offer service at competitive prices. Could lead to the return of charter flights operated by American or possible new UK owned long haul charter airlines I suppose, but it will potentially challenge their vertically integrated business models.