Why 2014-2015-2016 were the years when the number of discovered exoplanets increased the most?
The big increase is due to the Kepler spacecraft. Kepler is responsible for the majority of the discovered exoplanets (https://exoplanets.nasa.gov/keplerscience/), so its data releases will have a big effect on the planet’s discovered by year. It’s easy to see in the plot attached (From: https://link.springer.com/article/10.1007/s12145-020-00464-7 ) how the years you mentioned match the announcement of discovered exoplanets using Kepler’s and K2 (same spacecraft).
On why it is ~2014 and not 2010/2011/2012, from my understanding that's when the campaigns' data started being released (https://keplerscience.arc.nasa.gov/k2-data-release-notes.html#k2-campaign-1).