In layman's terms, why do we put the "time" in space-time? What does "time" mean in the term space-time? I am trying to understand why anything that travels through space, from a proton to massive galaxies, is always referred to as traveling in space-time, not just space. Does this have to do with time being relative?

I understand that 3d space and time together constitute a space-time.

Can someone illustrate with a scenario to point out the need to describe things in terms of space-time.
To answer your point about why "spacetime" as opposed to space and then time - photons (and other massless particles) travel through spacetime on "null geodesics" - in other words they do not experience time at all. But they are not in some separate class as particles at approach ever closer to the speed of light travel on paths through spacetime that are ever closer to null geodesics.

In other words the four dimensional geometry is inextricably linked and "space" and "time" are a unified whole.

In theory humans moving also experience this effect but the speed at which we move is far too slow for this ever to manifest itself to our senses. But we do see its impact in high precision timing - such as with GPS satellites.