Palm的意思是手掌，Oil/Grease someone's palm什麼意思呢？給機械上油可以使機械工作順暢，給手掌上油是為了行事順利，也就是賄賂某人的意思了。
To grease/oil someone's palm is to give someone money in the hope of gaining a favor or information. In the other word, to bribe someone.
看看下面例句：
Mr Green is in serious trouble today ashe tried to grease taxation official's palm. 格林先生今天有大麻煩了，他竟然試圖行賄稅務人員。