I've always read how Britan and France were just about ready to give up, with French troops committing mutinies.

At no point were the British troops prepared to give up; despite a dip in morale after the end of Passchendaele, and the fiasco at Cambrai in 1917, British morale remained stable. Alexander Watson's book Enduring the Great War is recommended reading here; when you did see largescale surrenders of British troops during the Spring Offensives in 1918, these took place after A) British troops were surrounded and ran out of ammunition to resist with; or B) British forces were simply overwhelmed without opportunity to resist. When given the chance to fight, they fought, and remained in solid morale until the armistice. As for the French, the 1917 were a momentary reaction to the failure of the Nivelle Offensive. Once Petain took over the Army, and guaranteed better food, more frequent leave, and better planned, limited objective operations (as /u/zebra_humbucker alludes to from Stone's book), morale improved, and the French Army was an essential player in the recovery of the BEF in 1918 and the final Hundred Days Offensives.

Why weren't the Germans able to break them?

They weren't able to break them, because they lacked the resources to do so. With regards to /u/zebra_humbucker's answer, the war wasn't really won by chance. The radical solutions sought to Germany's deteriorating position after 1916 by Hindenburg and Ludendorff accelerated Germany's isolation, by bringing the United States into the War whilst Russia collapsed into socialist Revolution in October, also due to German efforts. When the Germans attacked in the West in 1918, their vague goal was to restore maneuver warfare and cut the British off from the French, but American divisions allowed the allies to create new reserves and Germany's isolation from international markets left it with a declining industrial capacity and access to food and livestock. Civilian morale wavered while the Army lacked sufficient horse and motorized transport to effectively exploit it's breakthroughs with cavalry and to enable it's infantry divisions and artillery to follow-up speedily. By July 1918, Germany lacked the reserves to take the offensive operationally, and was at the mercy of the Allied armies in the west, whilst later offensives at Salonika, in Northern Italy and in Palestine and Mesopotamia brought the collapse of their allies.