Throwing in the Towel:
When you throw in the towel what you are doing is giving up. You are quitting because you have decided that you can’t take it any more. You are admitting surrender or defeat.
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.
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