Incite, urge ahead, provoke. This expression has nothing to do with hen's eggs but comes from an Old Norse word, eggja, "to edge." Both edge on and egg on were used interchangeably, but today the latter is preferred.
Example 1: Jack is always egging me on to drive faster.
Example 2: Seemingly quiet, Margo actually eggs on Donald to quarrel with his staff.
Example 3: He rings up bosses where there is a dispute and either eggs them on, or urges them to continue the dispute.
Example 4: Her sister Vanessa egged her on to drive a hard bargain and advised her to withhold the story unless 'money is paid beforehand'.