First, look at the first paragraph of the second page. It says:
The things that you can see at some time are the inside of the light-cone at o.
I don't get it. Suppose an event happened outside the light cone. Light from the event will reach me at some time, right? Because... Light keeps moving. For example, say, an event happened at (-1s, 5 Light second). So 1s in the past and 5 Ls away. This is clearly outside of my light cone at o. But I will see this event once light reaches me from there after 5s.
Next, the last paragraph of the second page. I only understand the first two lines there. I've no idea what this part is about:
As the horizontal plane hits the hyperboloid containing q, what you will see is a point that turns into an expanding sphere about you. As you watch the sphere expand, while the physical points are moving away from you, the spacetime distance between your birth and the sphere remains same. The increase in time is canceling out the increase in spatial length in the Hyperbolic Pythagorean Theorem, equation (1), since the separation is constant.
When the horizontal plane hits $q$, I should just see the event $q$ happening, right? What expanding sphere is it talking about? I don't get any of the next lines.
The text is from A Geometric Introduction to Spacetime and Special Relativity by William K. Ziemer.