How would increase labor productivity lead to improved living standards of people in an economy? I would like to know more about the mechanisms through which higher labor productivity would translate into higher living standards for a population in a given economy.
In economics we measure living standards by the amounts of goods and services people can consume. Increasing labor productivity allows us to produce more goods and services and thus also consume more goods and services, hence it improves the living standards. 

For example consider classic Cobb-Douglas production function:

$$ Y = K^{\alpha} (AL)^{1-\alpha}$$

in this production function $A$ can be viewed as a productivity of labor (or technology that makes labor more effective). In the equation above $Y$ is income or output (i.e. amount of goods and services) $K$ capital and $L$ labor supply. 

Even if everything stays constant (meaning economy does not have more capital or people dont work more hours and so L is also constant), but the labor productivity increases ($A$ goes up) the output of economy goes up as well. Higher output means that there are more goods and services for people to enjoy and in economics we would count that as an increase in (material) living standards.