Western leaders hope the spread of what will decrease support for terrorists?
The best answer to this is liberal democracy. Western leaders hope that the spread of liberal democracy will create a situation in which fewer people feel the desire to engage in terrorism against the West.
It is important to note that this does not simply mean that Western leaders believe that elections will bring about less terrorism. Instead, they believe in the need for liberal democracy. That is, they hope that a form of democracy will spread that (like our system) respects and promotes human rights such as the freedom of religion. They hope that the spread of such ideas will promote tolerance around the world. If this is accomplished, people will be less likely to hate one another on account of religious belief.
Western leaders also hope that the spread of things like capitalism and free trade will reduce terrorism. The idea here is that greater prosperity will reduce the desire to engage in terrorism. Even so, the major hope (as expressed in President George W. Bush’s rationale for invading Iraq) is that liberal democracy will spread and terrorism will wane.