How to determine if a binary relation R on M is a bisimulation?

[Dragonfall]

Let M be a system (possibly proper class-sized directed graph where the class of children of a given node is a set). A binary relation R on M is a BISIMULATION if R\subset R', where for a, b\in M

aR'b \Leftrightarrow ((\forall x \in a_M\exists y\in b_M xRy) \wedge (\forall x \in b_M\exists y\in a_M xRy))

where a_M is the set of children of a in M.

So let ~ be the relation on the class of all sets V such that a~b iff there exists an accessible pointed graph that is the picture of both a and b.

An accessible pointed graph is a directed graph with a distinguished node from where every other node is connected by a path.

Show that ~ is a bisimulation.

I'm not sure what bisimulation says. Can someone explain it in more familiar terms?

 

[Dragonfall]

The wording above (quoted from the book) is very bad. It took me a long time to understand that R is a bisimulation if R is contained in the ' operation.