Note: This answer deliberately covers the rules at their most basic, ignoring the multitude of options and exceptions to focus on the question at hand. For instance, the question doesn't indicate the creature possesses the universal monster ability grab—a discussion of which would easily double this answer's length—, so that's ignored below.
A typical humanoid creature that makes a combat maneuver check to attempt to start a grapple using one hand suffers a −4 penalty on the check. One benefit of the feat Grabbling Style eliminates this penalty. However, the feat Grabbing Style does not eliminate this penalty if a typical humanoid attempts to start a grapple using no hands. Also, technically, a non-humanoid—such as an outsider like an aasimar—does not usually suffer this penalty, and such a creature can attempt to start a grapple using even no hands at no penalty. (A GM may nonetheless rule—reasonably, in my opinion, although I disagree—that this −4 penalty applies to creatures of humanoid shape rather than only those of humanoid type; ask first.) Finally, a typical humanoid suffers this penalty only on the combat maneuver check it makes to attempt to start the grapple, not on the following turns' combat maneuver checks it makes to attempt to maintain the grapple (contrary to, for example, the benefit of feat Piercing Grapple).
In short, in addition to its other effects, the feat Grabbing Style allows a typical humanoid creature to make a combat maneuver check to attempt to start the grapple using one hand without suffering a −4 penalty on the check, while, for example, in its other hand it either carries a shield or wields a weapon.
Eliminating that penalty has no additional effect. For example, the penalty's absence does not allow a typical creature that's already taken on its turn a standard action to make one combat maneuver check to attempt to start the grapple against one foe to, on the same turn, make a second combat maneuver check to attempt to start the grapple against the same or another foe.
Grappling limits a typical creature's ability to attack
Upon successfully starting the grapple and while the grapple continues, a typical creature gains the condition grappled, and that condition prevents a typical creature from making attacks of opportunity. Further, on each turn after a typical creature starts the grapple, it must take one standard action to make one combat maneuver check to maintain the grapple against one foe. Hence, on the turn after starting the grapple, a typical creature must take one free action per foe to release any foes it's grappled before it can take its actions normally. For example, if a typical creature wants to take the full attack action, it must first release the creatures it's previously grappled. (Note that the typical creature's grappled foe usually has available a wider variety of actions that are beyond the scope of this answer.)
Grappling multiple creatures
A typical creature can neither start nor maintain the grapple against multiple foes. A typical creature must take one standard action to make one combat maneuver check to attempt to start the grapple against one foe. Then, on its next turn, a typical creature must either take a standard action to make a combat maneuver check to maintain the grapple against one foe or take a free action to release a foe against whom the creature has previously started a grapple.
The Grabbing Drag feat's reminder that its benefit can't be employed when a creature's grappling two foes is usually superfluous. And it appears something untoward happened to the feat Grabbing Master during development, as its unclear language and mechanics maybe imply a typical creature can now do better something it can't do, which is just as confusing as it sounds and makes the feat Grabbing Master an outlier not the norm. Paizo's admitted the Advanced Class Guide is an "all-time low point, editing-wise," so forget about the feat Grabbing Master and move on. (And, while forgetting things, also burn away the memory of the feat Hamatula Strike.)
So. Many. Exceptions. Notice I've used weasel words like typically and usually so far, like, a dozen times? Every time I have, there's something—a special ability, an archetype or two, a specific class feature, an obscure feat or two or three or four, a spell or two, a magic item or two, and more—that's an exception to the typical or the usual. Sometimes I don't even know the precise exception but blithely assume one exists because if Pathfinder can tinker with it, Pathfinder's probably tinkered, and an answer that tries to address every exception would take weeks. Try not to feel slighted if a favorite exception's omitted.
Considering playing a luchador, a kraken, or Cthulhu? Then this awesome blog post on grappling is pretty much required reading. It walks the reader nearly word-by-word through all of grappling's permutations. I highly recommend it. I am in no way associated with the author. It's just really good.