It means "I will do something (action X) that eliminates the need for you to do something (could be action X or a different action)" Literally, it suggests that the speaker is doing you a favor and it can be used in this way. You say to your roommate "I'm going out to buy groceries" and he says "Let me save you the effort. I already brought some takeout restaurant food home for dinner tonight." However, it can also be meant in a not so kind way. There can even be a bit of meanness or at least brusqueness. For example, let's say that you left work without permission and, in your absence, something bad happened (like an accident or a fire). You come back and start to make excuses to your boss but he already knows what you did and what the consequences were. As you start to tell your story explaining why it's not your fault, he might interrupt and say "Let me save us both the time and effort. I know what happened so don't bother lying to me."