These fares do require decent flexibility -- for tourist destinations, you won't be leaving on a Friday and returning on Sunday. The £4.49 flight between London and Romania is on a Wednesday, on a Friday it's £60-70.
Ryanair use the low fares for marketing, pretty successfully. They probably expect many people to add luggage (£10-40, depending on route), or buy food on the plane (£10+) or be disorganized and end up paying more.
In Western/Northern Europe they use out-of-the-way airports -- London Stansted is 47 minutes by the fastest train from London, or over an hour by car. They have a reputation for reducing the fees they pay the airport by threatening to move to alternative airports, unless they're given a very good deal. People walk to the plane, they don't use jet bridges. The whole plane is economy class, and the seats plasticy-fake-leather, which must speed up cleaning. At airports, they'll usually use the worst (farthest) gates. All the planes are the same, so maintenance is simplified and pilots easily moved around.
As to why the US doesn't have them -- I don't know. Norwegian is a budget airline, and has fares like New York to Paris for $175 one way (Tuesday, no luggage, no food!).
The European competition authorities are quite strong. This is one part of popular opposition to Europe -- many regulations for standardization or unlocking restricted markets gore national heros in protected markets (there are plenty of other reasons for the opposition -- let's not start a flame war in this thread).
In the US this standardization process happened over a century ago thanks to the railroads. Starting in the 1980s competition law was successively weakened. In addition the cartels were able to buy legislation they needed. Hence there is little competition in airfare or internet service compared to Europe. Large chains control the downtowns of most cities and suburbs.
Explanation is extreme variability of rates. A 1000 mile return ticket in Europe can be $20 and can be $500, with average about the same as in the U.S. at around $200, i.e. 10 cents a mile. It mostly happens because most flights are between countries with extremely different living standards, and some passengers are business travellers from rich countries to poor ones, others are seasonal laborers going the other way around. There are no U.S. states with per capita GDP difference as big as U.K. vs Romania, for example.
