It meant that at least one army, the first, but also quite probably the second, and maybe even endangering the severely weakened third army, faced the danger of encirclement and total defeat.
Supply lines were already strained heavily, French resistance became more staunch, the 6th French army guarding Paris more or less a surprise, and the third army being quite sick with plain illness. When Kluck stormed ahead with his first army, first too far to the South then disorganising the plan already and then turned West to attack what's in sight for him, the interlocking advance of German armies, guarding each others' flanks, was no longer interlocking, creating the gap.
The BEF moving into it almost unopposed had created the possibility of cutting off the first and second army. While the ensuing panic on the German side may have been a bit of an overreaction, completing this encirclement would have most probably ended the war a lot earlier.
(From: Sven Felix Kellerhoff: "Ein Befehl ließ den deutschen Kriegsplan scheitern", Welt, 09.09.2014.)
More detail in a slightly outdated Sebastian Haffner & Wolfgang Venohr: "Das Wunder an der Marne. Rekonstruktion der Entscheidungsschlacht des Erstens Weltkriegs". Luebbe: Bergisch Gladbach, 1982. Which is the base for a nicely illustrated documentary: "Generale – Anatomie der Marneschlacht" (On YouTubeOn YouTube). The dramatisation and dialogues shouldn't be taken too seriously, but the frontline visualisations will show you what you look for. For a more BEF-oriented outlook, of course the Wikipedia page: First Battle of the Marne is amply referenced.