To harvest the plant you can right-click it. Left-clicking will harvest the plant as well as destroy it. Harvesting Cyazint may yield Cyan Dye. Harvesting while destroying can also produce Cyazint Seed Bags. The harvest of the plant is determined by the plant's Gain statistic.
A good way to increase the amount of Cyan Dye it drops, how fast it grows and how resistant to trampling it is, is to cross breed Cyazint with Cyazint as this has the highest chance of getting a Cyazint. Collect the seed bags, scan them through a Cropnalyzer and then cross breed the two Cyazint seed bags that have the highest statistics with each other as this will often yield even higher stats. Then you can continuously breed the plants with higher stats together until you are satisfied. A more in-depth view of crop stats can be found on the Cross Breeding and GGR pages. The stats are referred to as Growth, Gain and Resistance.
Cyazint Growth Stages: