We all know and use many ways of saying things, but would they still make sense if translated into other languages?
Well, literal translation is rarely a good idea and if you are dealing with natively known sayings, it's even worse. As I am Italian, I'll give some examples from English into Italian. Feel free to comment and add some examples in your native language!
For example, the saying "a leopard cannot change its spots" if literally translated into Italian it means exactly that, a leopard can't change its spots. An Italian native speaker wouldn't understand the idiomatic meaning. If we say "il lupo perde il pelo ma non il vizio" ("the wolf loses its fur but not his habits") it is well understood by everybody in Italy. Old habits die hard is another way to express the same concept in English.
The same happens with the figurative sentence "a dog-eat-dog world". Literally a foreign person would think about a world in which there are dogs who eat other dogs. If we translate it as "morte tua, vita mia", usually pronounced in Latin, "mors tua, vita mea" (in English "your death, my life") it acquires a meaningful sense, not applicable to the English language.
These sayings always have a history. They were all born in ancient times thanks to something that was believed or happened at that time. For example "mad as a hatter" ,a well-known phrase also from an international point of view thanks to Lewis Carroll's book and the Walt Disney Production animated fiction film "Alice in Wonderland". In Italy we say "matto come un cavallo" ("mad as a horse" in English), this is because when a horse goes crazy it's impossible to calm him down. Why do the English use the word "hatter"? The reason is because in the Victorian Era the hatters used to make use of mercury during their job, a chemical which affected their nervous system (and probably the reason they used to become mad).
Even if we don't know the equivalent Italian phrase and we need to translate these particular sentences, there are some cases in which the Italian phrase is just like the English one. For example "cry over spilled milk" is "piangere sul latte versato" (when we do something wrong and we are sad about it). We say it probably because when the idiom originated, milk was a very expensive and rare product and if it fell down it was a shame.
Another one is "Don’t look a gift horse in the mouth" (like the Italians use to say, "a caval donato non si guarda in bocca"), this means that you don't have to criticise or try to find some flaw in something given to you as a gift; you just have to be grateful to that person, even if the gift is not valuable or if we don't like it. This saying was born because you can figure out how old a horse is just by seeing the condition of his teeth.
The way sayings are used frequently in many languages and are often understood by the majority, as we use to use them frequently during our informal speeches in order to explain in a figurative and catchy way a concept that would need a large vocabulary to be explained.