To act faithfully is to behave in a dependable and honorable way. A dog who waits faithfully outside your school until you get out of class is devoted and trustworthy.
The adverb faithfully describes an action or behavior that's faithful — in other words, it's steadfast, loyal and reliable. An actor's parents might faithfully attend every play he's in, no matter how small his role is. Your devout uncle might faithfully attend church every single Sunday. In both examples, a person is dependable. The earliest meaning of faithfully was "acting in a way full of (religious) faith," or being pious, especially as a Christian.