They are called machine learning websites. Other than the airline websites, the best example would be
Amazon, which gives you suggestions based on your previous purchase and search history.
The airline websites have algorithms / programs which learn and remember your search history. The incremental price increase when you search the same route/ dates multiple times is to give a sense of urgency. The airline website tracks your movements and if you search a route and dates multiple times without booking, the price increase is to trick your brain into thinking, omg I need to book NOW as if I dont, I might not get on this flight. The airline wants you to book with them, instead of with a competitor and will create programs on their websites to make sure you complete the sale.