These two episodes of Enterprise takes place, chronologically in the Mirror Universe, in between the Original Series episodes "The Tholian Web" and "Mirror, Mirror".
In "The Tholian Web", the Enterprise encounters a ship, the USS Defiant trapped in a "dimensional rift", which eventually disappears. Kirk is aboard the Defiant when this happens, and is trapped "between dimensions" until, eventually, Spock is able to rescue him. They are hampered by a race of beings called Tholians, who claim that region of space as their own.
In "Mirror, Mirror", we are introduced to the "mirror" universe, which is an alternative-history version of the main universe. It contains "opposite" versions of all of the characters (all the heroic characters are evil, Starfleet is warmongering, etc).
These two episodes of Enterprise connect those two episode of the Original Series. First, we see the Vulcan first contact event, and a much more violent and aggressive humanity, leading to the very war-like Starfleet of the mirror universe. Later, Archer wants to go on a mission to capture an advanced starship from the future -- this is actually the Defiant from the main universe, having been sent into the mirror universe and the past by whatever dimensional rift it got trapped in. This means going into Tholian space to get it.
The "other Archer" that Archer sees is just a hallucination. While on the Defiant, Archer and Sato read through the ship's records. Mirror Archer is shown main-universe Archer's service record, including how important main-universe Archer was in forming the Federation of Planets. Mirror Archer then begins to hallucinate this "other version", with the hallucination constantly berating him for being a failure.