How do economies grow? These days, we hear again and again about the so-called &quot;need&quot; for economic growth. But how do countries actually grow economically? That is, why/how does their GDP increase over time?
Related, what are policies that are believed by economists to interact the most with growth? And what are these channels of interaction? I'm not looking for dogmatic answers on this part, make sure that you state the causal chain as neutral as possible and link references when possible.
See also

What is the relevance of economic growth?
What is the Gross Domestic Product (GDP)?
GDP is a measure of economic activity. One typical way to look at the question is the way shown in another answer. Here I want to put some assumptions in place in order to give a bit more structure.

It is widely accepted that economic output is created through capital and labor, and facilitated by technology. For the sake of answering this question, we can assume that somehow we can aggregate all the decentralized small economic activities into one aggregate economic activity. For our economy then, GDP $Y$ is produced using labor $L$, capital $K$, and technology $A$, and some sort of production function $F$.

$$ Y = F(A, K,L)$$

This entails all ways that growth can happen, and I shall walk quickly through them.

Labor

When there is more work being done, we produce more. This includes both the intensive margin (each worker spending more time per week, month, year on on his job(s)), and the extensive margin (there being more workers, due to larger population). 

Note that it is unlikely to get persistent growth from the intensive margin - eventually people are working as much as they can. The extensive margin, however, can give you persistent growth. 

One can also think about $L$ as effective labor, in the sense that human capital also influences output. A stronger man can harvest more fields than a weak one, and a smart man can design more bridges. If workers get more skills that are useful in their economic activity, that will also stimulate growth.

Capital

When there is more machines, we can produce more, for the same amount of labor. $K$ in that equation entails both quantity and quality of machinery, similar to the human capital point. 

Capital is part of economic output. Hence, in order to create GDP, we need GDP. We also need people to actually decide that saving previously earned GDP and "transforming it into capital" is better than consuming it. 

Capital naturally depreciates. This is because technology may change, and hence old tools might be of less use, but also naturally due to fatigue etc. Hence, even if we don't want to grow, we still need to consistently devote a share of GDP towards investment. If we want positive growth, we need to invest further.

Technology

Technically, technology incorporates both $A$ and $F$ in the previous equation. If we have a large population, but only little capital, creating technologies that make more efficient use of labor, and need less capital, will increase growth. Regarding $A$, using technologies that allow us to produce at a higher level of efficiency we can produce a higher level of output for the same amount of inputs. 

So then, what are good policies/channels?

Increase investment

Higher investment means more capital, and hence more to produce. An example policies to increase investment are decreasing capital taxation, and anything that increases the return to capital.

Increase effective labor

We cannot get persistent growth from increasing the working hour per person, although this may be a valid policy in the short run. Increasing the labor force (more immigration, higher fertility) is a  second. Be cautioned, that that also means that the population gets a smaller share of the GDP, as we have more people to split the cake. 

Increasing human capital hence seems like the natural way. Policies that allow people to become better at their jobs will increase growth. That could be 


better education
training on the job
couching of unemployed


Increase technology

This is a no-brainer, and most likely the most important driver of economic growth over the past century. In effect, many people are worried that the time of big growth is over, just because the time of big inventions and technological booms appears to be over. Policies that incentivize research and development will increase growth. Such as


Patents (but they are a double edged sword) and anything else that will increase the returns to research
Better education, make "research" more attractive
Technological/research hubs, similar to silicon valley: Anything that facilitates research