This means to "try" to do something:

I've made an effort to watch what I eat a little more carefully in the past few weeks.

You usually use it when you haven't been successful doing something, or you don't think you're going to be successful. In the example above, the speaker say that he's going to "make an effort" to spend more time with his family. That might mean that he's not really sure whether he'll be able to do that.