Despite what the article says, a creature with the special ability improved grab does not also have the unique ability to simultaneously drag away a grappled foe
As mentioned, the Monster Manual (even the most recent 2012 premium edition) includes in the special ability improved grab a section saying that
When a creature gets a hold after an improved grab attack, it pulls the opponent into its space. This act does not provoke attacks of opportunity. It can even move (possibly carrying away the opponent), provided it can drag the opponent’s weight. (310)
Emphasis mine. That a creature can move after having grappled another creature is true as per the Player's Handbook rules on grapple, but
You [the grappler] can move half your speed (bringing all others engaged in the grapple with you) by winning an opposed grapple check. This requires a standard action, and you must beat all the other individual check results to move the grapple. (156)
The Monster Manual's mention of a creature being able to move after having used or while using the special ability improved grab can be safely ignored. It is mentioned in improved grab, but the rules for doing so are never made clear for the specific case of the special ability improved grab so, instead, the general rules for moving a creature during a grapple are used. It is likely just leftover text from rules that were considered and abandoned.
Just in case this is worrisome, there are at least two other places wherein similar unplayable rules persist in the text. I know that it's really difficult to let go of such troublesome artifacts, but the system isn't perfect, and letting some things go is okay. (See this question about the spell major image and this question about accidentally activating magic items.)
Those columns aren't errata and can't make changes to the text. They carry the same or less weight as the FAQ (and the FAQ's authority is dubious). Use them with caution, confirming that any strangeness you perceive in the columns also exists within the text. Then go with the text.
...And on the Rules Compendium
The Rules Compendium presents the special ability improved grab as follows:
When a creature gets a hold after an improved grab attack, it pulls the opponent into its space. This act doesn’t provoke attacks of opportunity. The creature has the option
to conduct the grapple normally or simply use the part of its body it used in the improved grab to hold the opponent. (101)
Thus, the RC omits the disturbing sentence entirely. (Note that the Rules Compendium is not without issues.)
"But I want monsters to do this!"
That's cool. From Sandstorm the general feat Scorpion's Grasp mimics much of the special ability improved grab, including giving the DM specific rules he can choose to use if he wants to implement the It can even move sentence as originally presented in the special ability improved grab:
You have the option to conduct the grapple normally, or you may hold a creature one or more sizes smaller than you with your off hand. If you choose to do the latter, you take a −20 penalty on grapple checks against that creature and you cannot deal damage with your grapple checks, but you are not considered grappled yourself. You don’t lose your Dexterity bonus to AC, you still threaten an area, and you can use remaining attacks against other opponents. While maintaining this latter type of hold, you can move normally (possibly carrying your opponent away), provided you can drag the opponent’s weight. (52-3)
Were I a player, however, I would want the DM to make such house rules clear before, for example, being attacked by an ankheg (which can, in the same turn, make such an off-hand grapple then burrow away, my character in its mandibles).1
1 The phrase house rules is not in any way used pejoratively here (nor do I use it pejoratively elsewhere ever), but, instead, as an indication that the DM must essentially tell the players, "We're deviating from the text a bit: When a creature has the special ability improved grab, we're using some of the rules from the feat Scorpion's Grasp to complement that special ability."