How do property rights encourage economic activity?

This is an important question because it is central to a capitalist economy. Property rights act as an incentive to economic activity because they motivate people to work harder and invest more.  Some examples of this will demonstrate the difference between economic activity that is generated by property rights, contrasted with situations in which there are no property rights.  

Let's first look at the situation in which a person owns or does not own a home.  The person who rents a home is going to do far less to improve that home than the person who owns the home because the renter has no property rights to act as an incentive to create more value.  The homeowner will add a deck, perhaps upgrade the plumbing or wiring, put in a new kitchen or bathroom, or finish off the basement. This is economic activity because it increases the value of the property and creates work and income for others.  The renter will not engage in any of these activities because the renter will not reap the benefit of the upgrades. It is the owner who would do so, thus the renter has no motivation to put money, time, or energy into an activity which would benefit someone else.

Next, let's look at the economic activity of the business world, imagining a person who owns a business, contrasted with a person who manages a business for someone else or who manages a business owned by the government. The person who manages a business for others is far less motivated to work hard and invest resources than the person who owns the company.  We are always much better motivated when we reap the rewards than we are when someone else reaps the rewards of our hard work.  The business owner will put money into the business, hiring more people, creating more product, or upgrading equipment. This is economic activity that is motivated by the owner's property rights, far more motivating than managing for others.  In countries with government-owned industries, for example, the gas and oil industry in Venezuela, managers and workers have only a paycheck or perhaps fear to motivate them, with little incentive to create value for a faceless government.

Intellectual property is another area in which property rights create significant incentive for economic activity.  Patents and copyrights are property rights, as much as individual ownership of a house or a business.  Computers and apps, films and music - all are property.  The incentive to create or invent rests largely on the idea that the product of one's intellect is protected as a property right, so that one will reap the financial rewards of the work involved.  There is joy in creation, to be sure, but that cannot feed a family or pay the doctor bills. People who wish to create must find an economic activity that accomplishes those, and it is the fact that there are property rights in intellectual creations and inventions that motivates this.  This is one reason that stealing movies and music is so dreadful.  When we do this, we are taking away people's motivation to create beauty in the world.  The property rights in these creations encourage people to create, which is a significant economic activity, adding value and putting people to work. 

Capitalism works well because of property rights.  People are incentivized to work hard, invest their capital, and put in more time because of the benefits of ownership.  This is what creates economic activity.  Without these benefits, people are often motivated to do only the bare minimum.