This is an interesting verse, because we're taught that the first commandment is to love God and the second is to love our neighbors. One lesson is probably that pointing out people's faults is not part of loving them... but then again the Cain and Abel story suggests that at least in some circumstances we *are* our brother's keeper, and depending on our responsibility to another person, there are probably circumstances where that is part of loving them... a parent or a bishop for instance might need to help someone see a fault before being able to help work through it.
I think what the mote/beam story is about is similar to the command to "first seek to obtain my word" (D&C 11:21). The idea is not to say that we shouldn't help people who have things in their eyes, but rather to make sure that we are applying lessons and commandments to ourselves first, and that we aren't asking other people to live something that we aren't willing to live, or actively living. ... It's telling us to work on not being hypocrites. Perhaps we all need that lesson, at least to some extent.
Today, let's work on avoiding hypocrisy and making sure that we are living the way that we ask other people to live, treating people the way we would prefer to be treated, and being humble enough to hear God's warnings when we're messing things up. :)