What drives the advancement towards ever faster cellular network speeds?
what is it that makes it possible for cellular networks to keep getting faster


Basically, good old Moore's law. 

The handset is only half the equation. More modern and powerful silicon does help in getting better channel quality, less noise, etc. However this can't go above the channel bandwidth as per Mr. Shannon.

A simple way to boost the bandwidth available to each user is therefore to slice the landscape into smaller cells. Directional antennas on top of towers slice the "round" cell into quarters, like an orange. 

Installing lots of micro/picocells everywhere in densely populated areas  means each base station only handles a smaller number of users. Less users per cell means more bandwidth per user. This is enabled by reducing the price of base station hardware (ie, cheap silicon, Moore's Law, and MMICs which integrate the RF bits on-chip). 

A smarter system also helps. For example, in GSM, even when you don't talk, your bandwidth time slot is reserved for you, which is wasteful.

An important thing is also the availability of these at a reasonable price:


Big FPGAs with truly insane computation power
Fast ADCs/DACs
Microwave ICs


These enable digital radio, and this is where the juicy bits are, like MIMO and adaptive antenna arrays with real-time beamforming and channel equalization, advanced (and adaptive) modulations, plus strong error-correction codes which require lots of computing power, etc.