The French expression casser les pieds à quelqu'un is a weird one, a true idiom that doesn't directly translate.
Phrased correctly, it means to annoy someone. This expression has evolved from casser la cervelle to casser les oreilles to casser les pieds, with the meaning of casser being more to crush than to break.
It is a very commonly-used expression in French.
Il me casse les pieds avec ses problèmes
He really annoys/bores me with his problems.
The idea behind casser les pieds is more annoyance than boredom. But it's used with both meanings.
Note that the construction requires an indirect object pronoun. This means the expression is used with words such as me, te, lui, nous, vous, and leur.
Idioms are tricky in every language. In French, it's not usual to say "break a leg" to wish good luck to someone, for example.
Casser les Pieds
This is an odd-looking idiom. If you say "casser les pieds à quelqu'un", it means to annoy/bore someone.
If you say "casser les pieds DE quelqu'un" it's physical, and it means you broke someone's feet.
On jouait au foot...Pierre a tiré dans le ballon en même temps que moi. Il m'a donné un grand coup de pied et il m'a cassé le pied.
We were playing soccer...Peter shot the ball at the same time as I did. He kicked me hard and broke my foot.
Pierre a passé la soirée à me raconter ses problèmes de coeur, et quand je lui ai dit d'arrêter, il est allé casser les pieds à quelqu'un d'autre.
Pierre spent the evening telling me his love problems, and when I told him to stop, he went on to annoy someone else.
Synonyms
There are a number of synonyms for this phase, including some very common vulgar options that appear in everyday French language and pop culture.
Boredom
S'ennuyer (very common)
S'ennuyer comme un rat mort, or like a dead rat, which means to be very bored. (Common expression)
Se faire chier (very common vulgar slang)
Annoyance
Ennuyer, agacer, exaspérer, importuner (quite formal) quelqu'un.
Casser les oreilles à quelqu'un meeans literally to break someone's ears, but this expression is mostly used when someone talks too much.
Faire chier quelqu'un (very common vulgar slang)