More plausible explanation - many people don't like children and want to live for themselves but in poor/developing countries they have to 'invest' into children so they will take care of parents later. In countries with more or less stable economy people can rely on own investments or state pensions to not face extreme poverty after retirement.
Both are unlikely explanations given inverse GDP per capita and birth rate correlation: https://en.wikipedia.org/wiki/Income_and_fertility
More plausible explanation - many people don't like children and want to live for themselves but in poor/developing countries they have to 'invest' into children so they will take care of parents later. In countries with more or less stable economy people can rely on own investments or state pensions to not face extreme poverty after retirement.