I notice that prices for airfare (esp. North America to overseas) almost double, if not triple for some routes, in June and July; I figure there are more families on vacation with kids out of school, but is that the only explanation? It seems strange that the entire world (I mean, the Northern Hemisphere at least) all goes on vacation at the same time. Are there other factors?
People like to travel in the warmer months, no point going to lovely places if it’s cold and rainy. There’s limited supply of airline seats, and demand goes way up, so prices naturally go up. And demand is rather inelastic, people need to travel when the kids are out of school so they can’t move trips to other times of the year.
Lots of people prefer to go on vacations when the weather is warm, even if they don’t have children.
One way to avoid the airline summer price hikes is to vacation during the shoulder seasons before school gets out and after kids go back. Airfares are cheaper and you don’t have to deal with so many people while you’re traveling. It’s a win win. I refuse to travel in the summer months for just those reasons. The weather may not be as nice as it is during July and August, but it’s usually not bad for the entire time. I was never afraid to pull my kids out of school to go on a vacation in mid September or late May.
What about the winter months, I like to travel to avoid the cold–
And I don’t see the main attraction of flying to hot places in the hottest months of the year, like Athens and Madrid; I would have thought that flights there would be cheaper in July, and more expensive in September.
Try looking for cheap flights to Florida or Hawaii in December and January.
I’ll take a wild guess that you don’t live in the U.K. 
The big reason the prices spike so much is that supply is heavily constrained. We’re already operating almost every possible flight between the US and Europe that the airspace and the airports can physically hold. And that’s in January.
If we had a way to put 50% more capacity over the Atlantic in July we’d love to do it. But we can’t. So 100% of the adjustment to make demand and supply align has to happen on the demand side by pricing the marginal traveler out of the market.
A world where “summer vacation” was not in any schools’ vocabularies and France didn’t close for August would have very different price dynamics.