When the right causes and conditions arise a dormant karma seed (tendency) can be activated in the mind. Usually triggered through sensing something agreeable or disagreeable. Such as a sight, sound, taste, smell, touch, thought, idea, memory.
One feels like or dislike for what is sensed, and an identification with it. Followed by a grasping of that kammic seed, which leads to attachment. It is this attachment (clinging) that makes it grow, get stronger and in time bear fruit according to its kind, which will be either wholesome or unwholesome.
If one can become aware of these karmic seeds and discern whether a seed (tendency of the mind) is wholesome or unwholesome. One can choose with wisdom to not grasp the kammic seeds that lead to greed, hatred, and delusion. As those are the ones that cause suffering.
Craving is not something that can be suppressed though. It arises naturally from pleasant or unpleasant feelings, which in turn arise from sense impressions (sights, sounds, tastes, smells, touch, thoughts, memories and ideas).
But the grasping, the clinging is something we can interfere with, and it doesn't have to be done in a macho warrior way, it can be done in a gentle way. When craving arises in the mind, one doesn't have to fight it or judge it or try to make it stop. One can simply allow it to be there in awareness, give it space, let it be; but leave it alone, choose not to grasp or follow it, not to identify with it, not to add any more to it. Allow it to just simply arise, and then cease on its own.
This is where meditation practise is helpful. it is easier to do this whilst one is anchored with a meditation object such as the breath, the body, the elements: earth, water, fire, air, space, the four primary colours, or beautiful emotions like loving-kindness. There are a number of different meditation objects one can use to anchor attention and bring composure and stillness to the mind. Meditation trains the invaluable skill of serene undistractability, which leads to steady composure and equanimity, enabling one to be able to place attention where one wants to and keep it there contentedly. Centred like this, one can allow things to arise and cease in awareness without being pulled away or disturbed by them.
Starved of attention and without the grasping/clinging, the kammic seed won't be able to take root in the mind and will wither and die. Then one experiences cessation, non-attachment and all that is left is peace.
But not grasping is difficult to do. It happens so fast and we do it on autopilot, we don't even know we are doing it much of the time. We are ignorant of the process, and we have lifetimes of conditioning, of deeply ingrained habits and tendencies to contend with. This is where the noble eightfold path comes in, a gradual training, that teaches one the skillset and lucid serenity needed to become a master of not grasping (-: