This is sort of the mirror image of the sixth precept. In a practical, narrow, prima facie sense, this is an admonition against egotistic self-worship, and as such, this is pretty good advice. People who are in love with themselves are hard to tolerate. This mind-set is irritating and wearisome to those in it's midst. If you want friends, it's a really good idea to minimize this behavior. No one likes it, no matter how much they may otherwise like you.
A deeper look at the precept attends to the fact that the real problem is that there's nothing there to praise. This gets to the very heart of the dharma. This notion that we all have of an individually-identifiable, separately-existing entity, the self, is really a mass social hallucination. When we look carefully at what we commonly regard as the self, we don't find anything there. Fully realized, this seeing is enlightenment itself.
Therefore, any time that one engages in praising self, one is lost in the delusion that the self exists. This delusion is responsible for all of the pain, sorrow and misery of human existence--all of it. A full explanation of that is beyond the scope of this essay, in fact, this realization is beyond the scope of any explanation at all. You have to realize this on your own. Having said that, it is possible to notice when one is deluded by self-hood, and this precept is one such example of a way to do that.
But beyond that, on a more practical level, the only way to praise yourself is by comparison to others (this is why people around someone praising themselves find it difficult to tolerate--they are being derided at the same time). This process of comparison is fundamentally flawed even if one accepts, for the sake of discussion, a self. When you compare yourself to others, you are comparing your insides, i.e., your internal experience, to their outsides, i.e., your interpretation of who they are. That's really apples to oranges. There's no good information there, no matter what such a comparison may yield. It's like comparing sheet music to a performance, or reading a menu to eating food.
Moreover, idealization of this illusion of self makes it that much more powerful of a delusion, and therefore that much more powerful a force to bring about the pain, sorrow and misery it carries with it. The high and mighty fall hard. The higher and mightier fall ever so much harder. Once you truly realize what is at stake here, i.e., your very happiness and sense of peace, you will eschew attention to the self whenever you notice it. The trick is making the connection. No amount of the temporary and false sense of pleasure derived from self-praise is worth the inevitable cost. It's a fools bargain, literally.
So, to the extent that a follower of the way seeks to see things as they truly are, and seeing the self truly isn't, a follower of the was does not praise self.
Next - A follower of the Way does not possess anything selfishly.