Society is made up of old people and young people. Old people didn’t start off old; they were once young: working, having families, and paying taxes. These young people gradually age until they are old and no longer able to work. When this happens, it is important for them to be fully replaced by a new generation of young people. These young people will in turn work, have families, and pay taxes, which go in part to supporting the elderly population which can no longer support itself.
However, this setup only works if the young are more or less equal in number to the elderly. If the number of workers becomes too few to support the number of elderly, then the whole system faces the danger of a catastrophic collapse. Witness the debate over Social Security.
The list of developed countries is taken from the U.N.’s list of countries with very high human development (page 213).
The Total Fertility Rates for these countries are taken from the CIA World Factbook for 2010. Check it online.