Roughly speaking, a vector is just an ordered row of numbers, where you can use an index i to indicate what column a number is in.

That's pretty much what a continuous function f(x) is also, except that you now use a continuous index x to indicate what "column" a number is in.

So you can treat a continuous function like a vector (ie. you can add it, take scalar products etc.). We think of vectors as things having direction. Since functions can be thought of like vectors, we can use that analogy to think that functions also have some sort of "direction". This is just an analogy, which is why the author of the article you quoted said "the concept of direction loses its ordinary meaning". There aren't really any consequences, because actually the analogy works in great detail and can be usefully exploited.

I would actually say it differently: A normal vector gives us the direction in "real" space. Thinking of the function as a vector gives us a sense of direction in "function" space.

There are some differences between finite and infinite dimensional "vector" spaces, but the above is a quick and dirty way to think about it.