The PCB shown here is a demo board, described in this TI application note.
Point is that all the components in the schematic (p.4) that are not on top of this design are on the bottom side – and that includes two sockets, one for power only, and one that seems to carry serial data.
Why the lines go from that bottom copper layer up to the visible side of the board and back down? Layout! Sometimes, it's impossible to connect two things without making two lines cross. So, you build a bridge by taking one trace to the other side of the board and then back after crossing the other trace.
For more info, you'd have to read that IC's data sheet, and the specification of the platform – someone certainly wrote down which pins on these (hidden) connectors do what, and the PCB designer had to live with that.
Generally, having a ground plane next to an unbalanced RF-carrying trace is a common technique – a coplanar waveguide. You're right, in this case, the majority of E-Field should definitely be happening between the two balanced traces, so that's not the motivation here.
If you, hower, look into the document linked above, you'll find this anotated photomask of the top layer:
The text further up claims that the Dipole is already impedance matched, so that basically leaves geometrically practical aspects (you need have some sort of feed towards the antenna, and you don't want to start making funky non-rectangular ground pours, and if possible, the ground should be parallel to the dipole), aesthetics, and the fact, that, at least a little bit, the traces and components close to the RF pins on the IC might interact, so it's usually desirable to have them surrounded by ground.