To give up or surrender; to admit defeat and stop trying.
This phrase comes from boxing, where a fighter's corner would literally throw a towel into the ring to signal that their boxer was conceding the match. The practice became formalized in boxing rules by the early 20th century, and the phrase quickly moved into general usage.
Boxing gave English many metaphors for life's struggles, but this one is particularly visual and dramatic - the image of a white towel flying through the air as a symbol of surrender is both vivid and final. Interestingly, the towel was chosen because it was readily available ringside for wiping down fighters.
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