As fatar760 said, the recommendation for singing or humming is because this gets your brain involved. It's what I meant in an earlier post that you need both hands and brain participating.
Also, getting the singing/humming part going is another step on the way to hand independence.
It's interesting that you mention Happy Birthday -- this is a song that most of us know so well, so it's not surprising that you feel like you could do that one.
It should also drive home the point that often, we don't know a piece (or a passage, chunk) as well as we think we do.
But knowledge relies on learning, and learning comes from doing. So, by singing or humming out loud you are giving your brain another "doing" activity, which contributes to learning and ultimately to knowledge.
Really well said, and explained much better than my attempt.
A little example came to mind as I was reading:
How often have you sat in a car and a song comes on the radio, a song you THINK you know really well. You start singing along and you get through it.
Then, later on, you try singing it again but without the song playing at the same time. Do you remember the lyrics? Do you remember which section comes next? Is there a note in the melody that, maybe, you usually gloss over but now you're not sure what that note is?
Like ShiroKuro says, we don't often know a piece as well as we think we do. Only when we strip it down to its fundamental parts do we start to learn it and take ownership of it.