If you see an occasional ant or spider in your home, you can typically kill it yourself and be done with it; however, if your home needs treatment for an actual infestation of any sort, you don't want to try handling this on your own. Even though there are traps and sprays you can buy for use when the home is actually infested, you might note a few reasons why these may not be sufficient and why it's good to leave this work in the hands of a professional.
Where you see pests
You may see pests in an area of your home other than where they actually nest and reproduce. Not destroying that nest or colony can mean that your home will remain infested, even after you kill those few bugs or rodents that make it to your kitchen or basement. A pest control expert will know where and how to look for nests or colonies and will ensure all pests are destroyed—not just those few that you do see.
Effectiveness of products
The products you buy at a home improvement store for fumigating your own home are usually much less effective than those used by a professional pest controller. They may have far fewer active poisons or weaker poisons than those used by professional fumigators; this is to ensure the safety of consumers who aren't accustomed to working with these materials. However, this also means they may not work to effectively kill the pests in your home! A professional fumigator will use products that work quickly and effectively and will also know what precautions are needed to ensure you are protected from those products as well.
Reason for infestation
Some homeowners make the mistake of just spraying bug killer or setting down traps to stop a current infestation, without pinpointing the reason for the infestation. However, if you don't find out why termites attacked your home in particular, how mice got into your home and so on, those pests will probably just return!
A professional pest controller should thoroughly inspect your home for cracks and openings that are used by pests, for standing water that pests use as a water source, for a lack of cleanliness around the home and for other such issues that can contribute to a pest infestation. Their advice can help you avoid a future infestation and help protect your home from more damage done by those pests.