A witness server is a normal PostgreSQL instance which is not part of the streaming replication cluster; its purpose is, if a failover situation occurs, to provide proof that the primary server itself is unavailable.
A typical use case for a witness server is a two-node streaming replication setup, where the primary and standby are in different locations (data centres). By creating a witness server in the same location (data centre) as the primary, if the primary becomes unavailable it's possible for the standby to decide whether it can promote itself without risking a "split brain" scenario: if it can't see either the witness or the primary server, it's likely there's a network-level interruption and it should not promote itself. If it can seen the witness but not the primary, this proves there is no network interruption and the primary itself is unavailable, and it can therefore promote itself (and ideally take action to fence the former primary).
Note: Never install a witness server on the same physical host as another node in the replication cluster managed by repmgr - it's essential the witness is not affected in any way by failure of another node.
For more complex replication scenarios,e.g. with multiple datacentres, it may be preferable to use location-based failover, which ensures that only nodes in the same location as the primary will ever be promotion candidates; see Handling network splits with repmgrd for more details.
Note: A witness server will only be useful if repmgrd is in use.