eg. He ate the cake quickly, In this sentence 'quickly' modifies 'ate' but they are not close to each other.
The adverb is close in that sentence. It is not immediately after, but in warnings against putting adverbs far from the word that they modify, the intent is to warn against lack of clarity, because the adverb can not be connected to the proper word by the reader.
Far means something more like
He ate the cake, which was chocolate with strawberry frosting and chocolate chips rather than the usual vanilla frosting with candy hearts, because it was made by his favorite bakery, down the street, quickly.