Some obstacles in the Odyssey can be outwitted cleanly. This one cannot. Between a six-headed monster and a ship-swallowing whirlpool, Odysseus faces a strait with no safe passage at all — only a calculated choice about which loss he can survive.
Circe, once again serving as Odysseus's guide through the epic's most dangerous stretches, gives him the only advice that makes sense of an impossible situation: sail closer to Scylla's side. Losing six men — one for each of her heads — to the monster is a certain, survivable loss. Drifting too close to Charybdis risks losing the entire ship and every remaining man at once. Odysseus follows the advice exactly, watching six of his crew taken by Scylla as he steers deliberately away from the whirlpool's greater danger.
Where the phrase comes from: this exact dilemma — forced to choose between two dangers with no safe third option — is the literal origin of the modern idiom "between a rock and a hard place," alongside the closely related "the lesser of two evils." Few myths on this site trace so directly and completely into everyday modern language.