More then likely it implies just another reason among many others. Such as in this example sentence- "The babies in this video are cute. Adding yet another reason why I want a child".

That's my guess.

"Enter reason #474 we wish we were our toddlers (being pushed in a stroller and having everyday nap-time are fighting for number one)"

The person is talking about the video of the toddler dancing, and says that "being able to dance without worrying what people think about you" is another reason why people might be jealous of their toddlers. They say "reason #474" just to imply that there are many many reasons that people might be jealous of toddlers. It probably isn't actually the 474th reason. :)

In the parentheses they say "(being pushed in a stroller and having everyday nap-time are fighting for number one)". They are saying two more reasons why someone might be jealous of their toddlers, and that these two reasons are "fighting for number one", meaning that they both might be the number 1 reason why we wish we were our toddlers.

I hope I wasn't too vague. Don't worry if paragraphs like this are difficult to understand, even I had to read through that sentence a few times before I understood it!

As your other responders have said, 474 is just a number chosen more or less at random to suggest that the list of reasons is very long. I'm surprised that "Enter" didn't give you any problems; just in case, it is a casual way of saying "I'm about to introduce reason #474..." or "you're about to hear yet another in a very long list of reasons we wish we were toddlers".

Thank you folks. I got it. :)
I guessed #474 might be a specific number implying something.
Actually it is too simple.

No one made any corrections, so I am. :P

What Does…
When I read the subject, I thought there would be a list of things and I'd have to look at the 474th one. Putting "Reason #474" in quotes makes it clear that you mean that exact phrase.

I just…
At first, I wanted to say "this article" because that's natural. But…you also had "this morning," so it turned into "I read this article this morning" which is okay but looks weird. @_@
So I just chopped it to "I just read this" because "morning" isn't important and of course you had to read it online since it has a video.
I also don't know if it counts as an "article." Technically, it IS "a piece of writing included with others in a newspaper, magazine, or other publication" but it's so short. =_=
Would a real editor allow a sentence like that in a real newspaper? I don't know. Do we have real editors around anymore?
Actually, that's the wrong way to use "everyday," so probably not!
Oh, HuffPo, how I dislike you...

So, what…
I don't think there are implications.