Scaling Adjustment Costs
The New Palgrave Dictionary of Economics Article on Adjustment Costs gives a nice overview of the use of adjustment costs in economic models. 

What is the purpose of scaling investment costs with K here?

This function form implies that adjustment costs of size X% of capital $I/k$ cost the same fraction of capital ($c(I/k)$) regardless of the amount of capital($k$) involved. It is a constant returns to scale form of adjustment cost. 

It seems that this gives the property that marginal cost of investment decreases in existing capital stock. 

I think you have the marginal cost of adjustment wrong here.

$$ \frac{\partial}{\partial K} K \cdot c\left(\frac{I}{K}\right)$$
$$ = c(\frac{I}{K})  + K \cdot c'(\frac{I}{K})$$

The first term is decreasing in $K$ for fixed $I$ but the second need not be.