I was recently reading on this, but found that a complete definition was vague.
What does a brain state mean?
A brain state is a snapshot of everything in the central-nervous-system.
A brain state is said to contain everything about a person right the instant it is snapshotted: memories, emotions, skills, opinions, knowledge, etc.
Most importantly it is a snapshot of who you are right that particular instant the snapshot is taken.
If you buy into material/reductionist theories of consciousness, then a brain state is by necessity information about the physical constituents of the brain. If not, then substitute for a snapshot of whatever you believe a person's memories and opinions are made of, in order to understand the thought-experiment.