Verified answer

Surface area requires two dimensions, volume requires three. I suppose it might help to realize that that very far from the origin, the horn may be accurately described as a very thin two dimensional strip, ignoring any third dimension entirely. Of course, the graph doesn't actually look like this, but it wouldn't matter if you said it did, because it's infinitely thin.

The Koch snowflake is a fractal that encloses a finite area within an infinite perimeter, and does so in a readily understandable way. It might help to look at this to understand Gabriele's horn. A three dimensional Koch Snowflake ("sphereflake") can also be created; I've got one on my desk (only the fifth iteration - a "real" sphereflake would be quite remarkable!)

By the way, I don't think there is any way to enclose infinite area within a finite perimeter, or infinite volume within a finite surface area. I'll eat my hat if you can proove this!

I don't know how to prove it for gabriel's horn, but analogously for the are under 1/x^2 from 1 to ∞, you could just take rectangles of width 1 and height 1/x^2 at their top left corner clearly the total area of the rectangles is finite since ∑1/x^2 from 1 to ∞ converges.

I don't know if theres a way to understand it intuitively. Its a fact, you just have to accept it.