what is the reason for this tapering, and how do you actually read the microwave signals from these ports?
This feature is used so that wires can be bonded to the transmission line. The end of the transmission line itself does not have enough area to reliably bond a wire to, so it opens up to a region of larger area (page 116 of Daniel Sank's thesis explicitly refers to this feature as a "bond pad").
The bonded wire is then brought to a connector, which passes the signal through a readout chain consisting of demodulators and filters. The actual architecture of the readout chain can differ depending on who designed it (see page 127 for one example), but will typically consist of demodulators, amplifiers, and filters that prepare the signal for measurement with an analog-to-digital converter.