What drives and/or allows the move to higher-bypass turbofans on jetliners?
For a clue look at the development in the price of oil:



Historical oil price chart (source)

While the first development into the 5:1 region predates the first oil crisis (1973) and was driven by fuel economy, better range and lower airport noise, the next step was kicked off by the second crisis in the early 1980s. It took one decade to develop the engines and the airplanes which could fit them, so the Boeing 777 - GE90 combination came out in 1995 in a time of falling oil prices. 

The next round was only kicked off after the US invasion of Iraq which made oil prices shoot up from 2003 on. Again it took a decade for the engines to become available, but now the increase in diameter is less dramatic and existing designs can be fitted with them (A320neo, 737MAX).

Before the early 1960s the bypass ratio could not grow more because


jet engines were developed for military aircraft, and more emphasis was placed on top speed than on efficiency. Airliners used what was available (note that the Convair 880 was powered by a simplified, non-afterburning version of the J-79, which powered the F-104 Starfighter, the F-4 Phantom II and the B-58 Hustler).
early airliners (De Havilland Comet, Tupolev 104) had their engines in the wing roots, so little space was available for an increased bypass ratio. When Rolls-Royce was developing the first turbofan engine, the Conway, they limited the bypass ratio to 0.25 so it could fit into the Comet and the HP Victor bomber, which also had its engines in the wing roots.


The first oil crisis of 1973 kicked off the development of the unducted propfan (UDF) engine, which has been held back by noise problems. Again, it took about a decade until it could take to the air.



GE36 UDF engine fitted to the MD-81 (picture source)

The development of a new engine takes longer than that of an aircraft, so every time a big push, triggered by a substantial jump in oil prices, was needed to start the development of a new generation with significantly higher bypass ratio. Once the engine technology was ready and certified, it would be quickly incorporated into many models, because the better fuel economy would force everyone to upgrade.