Why do you need to be 20-40 years old?

Schools are looking for candidates that are old enough and mature enough to handle the pressures of teaching abroad. Although there is no guarantee with age, and there is no reason that an 18 year old could not necessarily handle this situation, employers prefer that candidates be a certain age to help ensure the success of their employment both for them and for the schools. In regard to the over 40 issue, this preference is due to the longevity and energy of the candidate. He/she may be placed in a classroom alone with 30 students for 5 hours/day. Once again, people of a certain age are not a homogeneous group. However, they are more likely to suffer from exhaustion and will be less able to stay with the potential school/employer for many years, thus making them a less competitive candidate.
That being said, I know some amazing teachers over 40 who love their students and have energy and enthusiasm far above their 20-something peers.

Also worth noting, you don't have to be age 20-40 in order to take one of ITA's TEFL courses. You can be any age over 18 to qualify -- it's just that the bulk of teachers tend to fall in this age range. I don't feel fully comfortable responding on behalf of ITA regarding their reasons for including this in their description, but want to make it clear that you can absolutely apply for one of their courses if you're 18 - 100!

I think it's more of a guideline, rather than a fixed rule. In some countries, for example in China, where they can be pretty discriminating of age. However, I have worked in Ecuador and Vietnam and in all my joe have had co-workers over and well-over the age of 40. Don't let that sentence hold you back! There are many jobs waiting!

Ages for people teaching English abroad varies country by country, some are set by the country for a work visa, some are cultural norms. Here's an article to give some insight but there are people 18 to 88 teaching English in different countries.