The second sentence is not a complete sentence. I think it was written to show when each acts as singular. However, it can be written in a better manner as follows:

1: Each of the three cats chases birds.
2: Three cats each chase birds

The rule that I follow is as follows:
In the first sentence 'Each' is the subject of the sentence. Whenever 'each' acts as the subject, it is always singular. Note that 'cats' cannot be the subject because it appears in the prepositional phrase.

In the second sentence, 'each' appears after the subject 'cats'. Whenever 'each' appears after the subject, it does not affect the number of the subject. Thus, the verb - chase is plural so that it agrees with the plural subject cats.

I hope this helps.

For e-GMAT users, this concept about each acting as a Subject is explained in detail in the file SV- Always Singular Subjects