LIMBO. That's the storyline. Throughout the entire game, you only see the title (aside from credits) and nothing else. This means that LIMBO's storyline is entirely dependant on what you make of it.
This could seem dangerous as most AAA title games contain storylines that have had their edges rounded and polished so that the developers know exactly what you'll be getting. LIMBO doesn't.
As the storyline's up for interpretation, the best way to play it is in one sitting. It's a fairly short game so a long afternoon's gaming session should see you completing it from start to finish, but doing so allows you to immerse yourself and fully experience LIMBO.
The desolate landscape and general solitude of LIMBO are the biggest influences on the storyline. Throughout the game you'll only encounter a handful of other humans, be they friend, foe or even lifeless.
The game's ending is also marvellous. For me, this was when everything suddenly clicked into place and I understood it all. Others, however, are rather more confused as illustrated by the fact that Google auto completes "LIMBO ending WTF".