When a tomato plant is pruned properly, all of the foliage receives adequate sunlight, and the plant is able to photosynthesize (and, as a result, grow and produce fruit) more efficiently.
But not all types of tomatoes need to be pruned. If you are growing determinate tomatoes, you don't want to prune. The reason for this is that determinate plants develop all of their fruit at one time, so if you prune, you're sacrificing tomatoes for no reason. But if you're growing indeterminate tomatoes, which produce fruit regularly over the course of a season, pruning is essential. Not only will it help keep these often-huge vines in control, but energy will go to producing several really nice-sized tomatoes instead of a bunch of smaller tomatoes (along with lots of foliage!)
Of course, tomato pruning isn't a required chore, no matter which type of tomato you're growing. If you're not overly worried about growing large fruit, or trying to keep the plants under control, you don't need to worry about pruning.
How to Prune Tomatoes
If you decide to prune, it's really a very simple process. What you are looking for are tomato "suckers," which grow in the "V" space between the main stem and the branches on your tomato plant. The first photo in this article shows what you'd be looking for. If left to grow, this would eventually grow into a full-sized branch--adding lots of foliage and, eventually, a few fruits. It will also result in a tomato plant that quickly outgrows its space in the garden.
So, to prune, you simply remove these "suckers." This can be done when they are smaller than two inches tall by simply pinching them off with your fingers. If they are larger than two inches, be sure to use a pair of clean pruners that you disinfect as you move from plant to plant to protect against spreading diseases. These should be pruned out when they are small to avoid stressing the plant by removing large amounts of foliage at once.
It really is that simple. It's a chore that can be done while watering or weeding, and one that will result in healthier plants and bigger fruits with very little effort.