The link between better education and economic growth comes from the fact that better education increases the amount of human capital that is available in an economy.
Economies can only grow by obtaining more or better resources. An economy can grow, for example, if it discovers new deposits of minerals that can be exploited. It can grow if its labor force grows, perhaps by something like immigration or women entering the workforce. It can grow if technology improves, allowing the economy to work more productively. In addition, it can grow if its workers are better educated.
Human capital refers to the skills and training that workers have. Like technology or minerals, human capital is a resource that a country can use. When people in a country get better educations, they have more human capital. They are able to do jobs that are more complicated and which bring more money into the economy. This is the difference, for example, between a person who designs new machinery and a person who operates the machinery. The former is much more valuable to an economy.
Thus, better education leads to more human capital which gives an economy more resources to use. With more resources, economic growth becomes possible. So the link between education and economic growth is that it is one of the limited number of options that allow for economic growth, and it is the most expandable option because educational opportunities can theoretically be expanded to include all.