The main reason for such trends are that the people from the developed nations tend to focus more on their careers and form families later in life, while in the developing countries it is the other way around.
Explanation:
In the developing countries, the population is increasing, while the developed countries it either stagnates or it is decreasing. There are many reasons that contribute to this, but the two main ones are the pursuing of career and forming a family later in life, or in the other case lacking options for career and forming family early in life.
In the developed countries, the people have much more options in life, so while they finish with the education they are already in their mid twenties, and while they progress on the labor market and reach a career goal they are usually in the mid to late thirties. This means that they often delay the formation of family and having children, so even if they want to they don't really have enough time to have lot of children, so usually have only one or two.
In the developing countries it is the opposite. The people are poor, usually have low education, no real prospects for career and instead work in agriculture. They tend to get married very young, even before their twenties, so they have much more children both because they have time on their side for it and because they need them as labor force and to contribute financially once they grow up a bit.