I really really want to understand this, but I can't get past your third sentence. No fault of your own, just me. But if you have anything I can read that explains it better I would be grateful!
The length of the red arrow is always the same. When your velocity is zero, your "time speed" is 1.0. As you begin to move faster, the arrow tilts towards the X axis, decreasing your "time speed".
The biggest issue is that the interchange between speed and time is not linear. So that graph doesn't make sense as displayed.
At a minimum you have to understand that the time scale on the side of the graph is not linear, but rather exponential. And graphs with exponential scales are not easy to understand.
Making it worse is that there are other things that can slow down time - namely acceleration.
I was mostly responding to TomGullen's confusion about the GP's statement, rather than trying to paint a super-accurate picture.
This is the same kind of thing, except that when "at rest" in space (relative to some fixed point) you're moving at the speed of light in a pure time direction. When you move in space, you divert some of that speed from the time direction into the space direction you're moving in. Overall you're still doing the speed of light, just in a different "direction".
"When you "move" through space, you're not changing the magnitude of your velocity vector, only its direction. So the more you move through space, the less you move through time."
I can understand what is being said here word-by-word, but the concept as a whole does not make much sense to me.