What does it mean?
The Buddha taught that everything is impermanent (anicca), full of suffering (dukkha), and lacks essence (anatta).
Why does it matter?
The first thing you have to understand is what Buddha meant by “everything”. He wasn’t (necessarily) talking about the planetoid Pluto. He just meant that all of our experiences have these characteristics. What else really matters anyway?
I think it is fairly easy for us to be ok with the idea that everything is impermanent. Not that it’s so easy to really live your life that way. If you really internalized that everything you love will someday die, you can see why all experience has suffering imbedded in it. Even the things that you love and often make you happy have the potential to bring the greatest suffering.
The one that is most difficult for most people is anatta. It is usually translated as “no soul” or “no self” or “not self”. I prefer no essence, because it gets at the connection to impermanence. I agree with Richard Gombrich here that the confusion comes because anatta is a shorthand for something a little more complex. There is an idea that you sometimes here that we are all waves on an ocean. Each consciousness is really just a manifestation of the universal consciousness. This idea is exactly what the Buddha was teaching against in the term anatta. The Buddha taught that there is no eternal soul, universal or otherwise. Our minds are compounded things. There is no underlying essence. In fact, there is nothing in this world with an unchanging essence.