Imagine a disc in vertical plane (this disc has markings on it which denote angles). Now draw a vertical axis along its diameter.. this is the "vertical axis" they're referring to.
Note that a magnetic needle is pivoted at its center.
Now say you're at some latitude (imagine a globe, okay?).. now you have infinite planes to align the disc to (about the vertical axis) (vertical meaning vertical w.r.t. ground at that latitude).
Now draw the perpendicular and horizontal components at your latitude. There will be exactly one plane where both of these components lie along the plane of the disc (This plane is the one containing the points: North pole, South pole, and your position).
Now flip this disc 90 degrees about the vertical axis mentioned. So now, the horizontal component is perpendicular to the disc (seeing that?).
Now the needle ought to point along the net magnetic field.. however due to obvious reasons, it is restricted to rotate on the disc only, so it will point along the only component of mag. field. This component is the vertical one, so it points 90 degrees upwards.
I didn't have much time to think this through.. so the language might sound ambiguous, so feel free to ask.