Essentially, it enables you to lock up all your passwords under one master password, which you'll be asked to create once you sign up. Pick something non-dictionary based (numbers and symbols are your friends, here). If you're setting things up via the mobile app, you can also add an extra layer of security by requiring Touch ID verification for access.
Next, begin adding all your accounts -- this will securely encrypt and save them on LastPass's servers. Each time you click to add a new one, it asks if you'd like to generate a new password for said account -- do this! Especially if you're the type who's been living dangerously by using a single password for everything. Their auto-generator allows you to create seriously tough-to-hack options -- you can even automatically copy them, which makes resetting them at the source site incredibly simple.
There are also a bunch of LastPass browser extensions. Download them! This makes it incredibly simple to quickly grab your login and password info without touching your phone.