Probably the most common kind of transfer among languages is loanwords: think of the many many foreign loanwords in English for example. It's easy to see how this happens. You learn a foreign word that has some convenient meaning, there's no good equivalent in your native language, so you start using the foreign word if you think the person you're talking to will know it too.
But in the Altaic languages, you see similarities in abstract structure, without much apparent sharing of words. Altaic languages have a certain strictly head-final syntactic structure, where the verb is the final element in the sentence, you have postpositions coming after nouns, lots of freedom to drop and reorder arguments of verbs, etc. They all have agglutinative morphology with case marking that creates long complex words. They all have, or show historical traces of, some kind of vowel harmony.
All these abstract commonalities happen with only very little sharing of words. You can line up a Korean sentence with its Japanese translation almost morpheme-by-morpheme, and yet those morphemes are all totally unrelated to each other.
It seems really odd that you could have a set of languages that share their abstract structure without sharing content. And Altaic isn't the only example of this kind of thing: you also see it in the North American native languages, which share a strongly head-initial head-marking structure, nearly the exact opposite of the Altaic languages, despite a lack of cognates among their morphemes. It's almost as if what was transferred among the languages was an aesthetic, rather than individual words.
We don't really understand how this happens. My best guess is that it happens in situations of massive multilingualism, so that certain habits of articulation and sentence formation get shared across languages within the mind of a multilingual speaker. It's something I'd like to understand more.