In Pre-Production, their job is to find the locations and establish all agreements. If it is an institution, they would need to go to the person in charge and ask for the permission to film there. If it is a private house, they need to talk to the owner and neighbors for permission to film. The owner and the neighbors may agree to let you use their house for free, or for a fee. If the neighbors don't like the noise level during your production, they might call the police on you, and you'll get booted out. So it's important to build a tolerating relationship from the neighbors.
They need to discuss the specific spots at the property, that will be used during production.
If it's a public place, or a location in the wilderness, things are simpler as far as obtaining permissions go. But the location manager needs to plan out how the location will be used, where the equipment will be placed, etc.
Once called upon by producers, they present the options available that would meet producer's needs.
One thing that makes it easier to film in a place like LA, is that there is already an established set of locations available for filming. There are a list of people, for example, who agreed to allow filming at their place. The producers just look through the list of available addresses and choose the one that fits the most.
During production, the location manager is working with the safety officer, to make sure everybody involved in making a movie, is safe.
They also watch to make sure nobody brakes anything at the property. Because at the end of the day, they will be the one to explain what happened to the owner.
Location manager is the first person to show up on set, and the last person to leave, because they need to make sure everyone is safe, and nothing gets broken.
Sources:
"Film Production Basics - Info for Anyone new to the Film Set Experience", published on August 17, 2012, http://www.youtube.com/watch?v=BHQjkWJJPgU