“Over 150 years ago, Darwin chose the tree of life as a metaphor for one of the most powerful ideas in biology: the relatedness of all living things. The power of that idea can be seen today in the ubiquity of evolutionary trees (also called phylogenetic trees) in all biological disciplines, from studies of newly discovered species to cutting-edge cancer research. To understand modern biology, we all need to understand how evolutionary trees can be read and used.” (http://evolution.berkeley.edu/evolibrary/article/0_0_0/evotrees_intro)
In this Tree of Life, time flows from the earliest at the bottom to the most recent at the top, as indicated by the arrow on the right. The branches represent all the groups of organisms that lived in the past. The groups listed at the top are those alive today.
The numbers on the branches, as explained in the key on the bottom, indicate that a particular trait evolved and was inherited by all the organisms on that branch above that point. All the animals on that branch have an ancestor in common that had that particular trait.
Try looking at one trait and follow the branches above it to see which groups have that trait. For example, #8: braincase and backbone. The place on the branch where the trait appears indicates the evolutionary root of that trait. No animal before that time has this trait; every animal on the branch after does. All the animals to the left of the lamprey, including the lamprey have a braincase and backbone. And none of the organisms to the right do.
Note that there was a diversity of organisms that evolved before the first animal, the sponge.
This Tree of Life was developed at UC Berkeley Museum of Paleontology and their website has lots of examples of and activities about this Tree of Life and Trees of Life, in general: http://evolution.berkeley.edu/evolibrary/article/0_0_0/evotrees_intro
And this website explains how to read the relationships on a Tree: http://creaturecast.org/archives/2820-creaturecast-phylogenies