This idiom means “to cultivate one’s character by studying hard” or “to be in a state of friendly rivalry”. This is one of the many 四字熟語(よじじゅくご- four character idiomatic compounds) that were assimilated into the Japanese language.

To better understand this idiom, it can be broken up into two parts:

[Jp: せっさ, Ch: qie1 cuo1]
to polish stones; to polish character; to learn from each other

[Jp: たくま, Ch: zhuo2 mo2]
carve and polish (jewels); to improve/refine/cultivate

1.ロシアの選手たちは切磋琢磨し、お互いから学んでいる / The Russian players work hard and learn from each other. (
2.ほとんどの台湾人にとって、いつの日かこの井戸を飛び越えて台湾の外の世界を見るために、切磋琢磨して向上することが非常に大切なのです。For the majority of Taiwanese, it is important to excel and advance so one day they can leap out of the well and see the world outside of Taiwan. (Taiwan A to Z: The Essential Culture Guide)