An idiomatic expression is an expression whose meaning cannot be inferred from the meanings of the words that make it up. It is an expression that completely defies the rules and definitions of all of its grammatical and vocabulary components, and thus is one of the hardest parts of a language for a non-native speaker to learn.
One of the best examples of an idiomatic expression in the English language would be "get up". From the point of view of a native English speaker, "get up" is a command that tells you to stand. Think of it from the point of view of a non-native English speaker, though. Taken from its parts, which is how most non-native speakers would take it, the command is telling them to GET, as in "take" or "grasp", the direction "up". To the non-native speaker, this makes about as much sense as being told to "Hold south" or "Be west" would to you. As several of these mount upon the poor language student, they can become very confusing, because they deny all logic, as well as everything else that the student has been taught in the class.
The solution to the problem of the idiomatic expression has been, for the most part, for the language teacher to simply tell their students something along the lines of, "Trust me. Just write it down, memorize it, and try not to think about what the words really mean."