Aside from the obvious benefit of free plants, there are several reasons why you would want to harvest your own seeds. It may be for sentimental reasons - a favourite family plant for example or because it is familiar and you are knowledgeable of its habitat and performance. Or maybe you are on a different crusade.

With the industrialisation of farming in the 1960s, it is estimated that 97% of the world's heritage seed has been or will be soon extinct. For many gardeners nowadays harvesting seed is about something infinitely greater than free plants. It is their personal contribution to maintaining and preserving the critical genetic diversity of the planet. Many people think this is so important they have chosen to make it their life's work. So before you begin harvesting seed consider planting some endangered heirloom varieties to harvest from - BECAUSE YOU CAN.
In England you can only grow seed that is on the 'Official List" and throughout the rest of the world the top ten seed companies, Monsanto being the largest, control half of the world seed sales under patents. Ironic perhaps that most of our heirloom varieties were born in the UK. 
So now you have selected your plants (hopefully heirloom varieties) how do you harvest from them?
Firstly select a good plant specimen and a dry day to harvest the seed. This will ensure the seeds are less likely to mould over winter. A tour of your garden should identify which plants are at the end of their growing season and which seeds can be collected. Cut the stems of the plants with the seeds on them and place them into a big bowl or alternatively place a plastic bag over the stem and turn it upside down after cutting the stem to catch seeds. This technique is especially applicable if the plant is very dry and at the end of seeding. When you have harvested all of your seeds allow them to dry indoors at room temperature on a flat surface. Be sure to turn them every few days so that the seeds dry thoroughly, evenly, and naturally. Some people chose to use desiccants to keep seeds dry such as silica gel packets that come with new electronics or they make their own - a teaspoon of powdered milk in an envelope or paper towel will also absorb moisture.
I use old recycled envelopes to store seeds after drying. I date the packet and state where the seed was harvested from. Then I place the packets into a moisture-proof container inside my freezer or refrigerator to be used within a one-year period to ensure maximum potency and germination.