Let me share what’s going on.
So, as @lemmings pointed out, we’re using a “ground plane” to connect all the ground points - basically, filling in all the blanks. When you load the brd file, it’ll look like nothing is connected:
but you’ll also note there are no yellow ‘airwires’ suggesting they need to be, and if you run a “Design Rules Check” it won’t throw errors about ground.
There are two reasons for this; firstly, I’ve turned off airwires for the GND signal. And secondly, a clue: those red/blue dotted lines around the edge: that’s a polygon that hasn’t been filled in.
If you click this icon in the toolbar:

or type RATSNEST; in the command line (if you’re me), this will happen:
and you can see that all the copper has been filled for the polygon, surrounding all the pads at a reasonable distance, but going through the GND pads. (The cross shape you can see is called a ‘thermal cutout’ - basically, it stops heat dissipating enough so you can solder it; if the GND pads were just connected entirely to the solid copper, it’d take ages to get them hot enough to solder GNDs).
In short: the plane is there, but EAGLE never fills them by default on load.