"Throw up the sponge" has the same meaning as today's more common phrase "throw in the towel." As Lewis says, it means to give in, to quit.
According to fun-with-words.com, "In its original form, to throw up the sponge, ... appears in 'The Slang Dictionary' (1860). The reference is to the sponges used to cleanse combatants' faces at prize fights. One contestant's manager throwing in the sponge would signal that as that side had had enough the sponge was no longer required. In recent years, towels have been substituted for sponges at fights, and consequently in the expression too."