How is an increasing population good? What are its advantages in the field of economics? What are the innovative ideas or fields in which...

Population growth carries both costs and benefits, but based on decades of research and debate, the consensus among economists is that a moderate level of population growth is better than either a very high rate of growth or a very low one.

Larger populations have a number of benefits.

Obviously, there are more workers to get things done, but it's not obvious that this would increase the wealth per person; yet typically it does, because economies of scale emerge. A lot of goods can be produced more efficiently with more people, at least up to a certain point. This is especially common in high-tech or capital-intensive industries. (There's a reason the United States is the world's leading producer of airplanes, and not Luxembourg.)

More people also means more specialization; we're already quite specialized, yet over time we have become even more so. At one time there were "scientists"; then there were "biologists"; now we have "computational biophysicists" and "bionanoengineers". Specialization allows people to work to their comparative advantage and engage in "trade" with other specialized people, producing at much greater efficiency overall.

Population growth is also linked to innovation; there are many other factors affecting technological innovation, but simply having more people means having more brains to potentially come up with good ideas, and more opportunities for those brains to interact with one another and support each other. In large cities, "tech clusters" tend to emerge for this reason; the benefits of having good ideas are amplified by being surrounded by more other people who are easily accessible and also have good ideas. You mentioned agriculture and nanotechology specifically, and this could certainly apply to those, but really it applies to just about everything. People get better at doing stuff when they bounce ideas off other people.

There are of course costs of population growth; however, more people means more mouths to feed, and if you are constrained by living space, diseconomies of scale, or natural resources, you might not be able to produce as much per person as you did when you had fewer people. The faster the growth is, the harder it will be for society to adjust and efficiently employ all those new people.

There are also demographic effects of very low or very high population growth. If growth is very slow, you tend to get a lot of old people, and thus the ratio of retired people to working people can decrease, putting a strain on pension systems. But if growth is very fast, you get too many young people, and instead the strain is on your educational system and entry-level employment. You want to have a balanced age distribution, and that tends to be achieved with moderate levels of population growth.