The ‘why’ you prune out the suckers on a tomato plant is relatively straightforward, they take away energy from the main tomato plant that could be better utilised elsewhere. So by pruning out tomato plant suckers you are helping the plant flourish and produce more fruit that ripens.
The term ‘pruning tomato plant suckers’ is a little bit of a mis-representation because what you are really doing is pinching out the suckers with your thumb and forefinger. You just take a firm grip on the sucker, twist and pull removing the sucker from the stem of the tomato plant. A good time to do this is when the plant is still quite young and when you are putting in some stakes or other form of tomato growing support. If you prefer to actually see a demonstration of how to prune tomato plant suckers then have a look at the video below. It is a very quick look at the technique as demonstrated by Doug Green.