Why do poor countries usually have high birth rate and rich countries have low birth rate. Isn't population supposed to be useful for the economy because they have bigger labor force?

Answer 1

It has to do with several factors:

• the most valuable work (the work other people will pay most for) tends to be specialized and requires years of education and/or training
• with plenty of people in the world, increasing efficiency and more machinery, the value of unskilled labor - just two empty hands - is not very high, and may actually be decreasing
• with an upper limit on Natural Resources, fewer people in an area means more resources per person

The lower Birth Rate in richer countries can be both cause and effect:

Cause: Children are expensive, more children even more expensive. With fewer children, you can spend more on each child, for example on education, and still have more for yourself.
Someone I know grew up in a rich country as part of a group that believes in big families. They were more than a dozen people on one income, and were very poor. It should be no mystery that feeding twelve people on one income will make a poorer family than feeding four people on two incomes (the current family average in 'rich' countries). That is, too many children can make you poor, even if you weren't from the start.

Effect: When large parts of the population are busy with education until their early twenties, it delays the time they start thinking about family. Career plans may delay it even more.
Generally, people have fewer children the more education they have. (Even within the same country.)