People say this when they feel like travelling without planning in advance. If someone travels toward the horizon or rising sun in the east, they might not have a specific destination. If they "head out for the horizon", they travel "where the road leads them" or even "where their nose leads them", they are going without much preparation and just being spontaneous!

A "heading" can be a direction. A ship might be "headed north." To "head out" carries an idea of space, expansiveness, big distances. At the same time, it carries an idea of "outward" from some reference point. "Where are you going this afternoon?" "I'm heading for the supermarket" (a short distance). "Where are you going for your vacation?" "We're heading out west to Yellowstone National Park" (a long distance). The "horizon" is the line that separates the sky from the land or the sea. If it is not blocked by trees or buildings, it is a full circle. You see the horizon in every direction, all 360°. So it carries the idea of "no direction at all." The horizon is also the farthest distance you can see. Furthermore, you can never reach the horizon. No matter how far you get, the horizon is always ahead of you. So it carries the idea of big distances, endless distances, endless travel. To "head out for the horizon" is sort of a contradiction. It is a colorful way of saying you are not heading in any direction at all. Any direction will do, it's all the same to you, you just want to go far, far away. It is a poetic way of describing the craving for adventure, traveling, always seeing new things, never staying put. To say you are heading for the horizon is a way of saying you are never going to stop traveling, because you can never reach the horizon.