Should every instance of application have their own cassandra session or they should share the same session?
You share your session on application level. Different applications should have own sessions.
Your quote means that you dont open a session for a query but rather have a singleton Session instance in your application.
I have a cluster setup of 3 nodes. 
I am designing my microservice and I am wondering if each node should have their own Cassandra session or if all three should share the same session created by any of the nodes.
I have read in the Cassandra docs:

"The Session instance is a long-lived object and it should not be used
  in a request/response short-lived fashion. Basically you will want to
  share the same cluster and session instances across your application."

What does this mean?