Is the labor force participation rate related more to labour supply or to labour demand? Or both?
Traditionally, labor force participation is best thought of as a supply-side phenomena. That is, a certain subset of our nation's population is jobless but is jobless by choice. For example, consider a stay-at-home parent. So, we have agents making a decision to not supply labor. 

However, there is a hint of the demand side at play here. For example, a very weak demand side might discourage workers, thereby compelling the marginal participant to exit the labor force.  

So, to be explicit: LFP is a supply-side concept. However, there is a mechanism through which demand side forces can factor into LFP.