To "curry favor with" someone means to try to gain their favor or approval through flattery, praise, or other means, often with the hope of receiving some benefits or advantages in return.
It involves ingratiating oneself with another person, typically in a sycophantic or insincere manner, to win their favor or goodwill.
For example:
"He was always currying favor with the boss by complimenting her and offering to do extra work."
"The politician tried to curry favor with the voters by making promises he couldn't keep."
"She constantly curried favor with her wealthy relatives in the hope of inheriting their fortune."
The phrase implies that the person is attempting to manipulate or gain an advantage from the person they are trying to please, often for personal gain.