How does square metal plate antenna work?
Andy's comment is correct- this is a microstrip patch antenna, which has a hemispherical radiation pattern. 



The construction is similar to a capacitor with a ground plane on the back, a dielectric (possibly ceramic in this case), and a rectangular patch on the top. It's designed to resonate at the desired frequency (1.575 GHz in this case). 

The round bit is the inner feed to the patch, which is impedance matched to the receiver input through a network (a network which in microwave-world is just a pattern on the substrate). 

Here is a calculator to find the dimensions, given \$\epsilon_r\$, dielectric thickness, and desired resonant frequency, and here is some info from Rogers (laminate maker) on designing your own.