Your dog bit someone, not on purpose. But tonight he is at the animal shelter anyway. He's in a cage, and he's upset. The animal control officer has told you that you will have to pay for boarding the dog in order to get him back. This imprisonment is going to last 10 days and cost you money.
You are angry! At yourself, for not preventing the incident, and at the "system" that has locked him up. What gives them the right to do that?
It's for your protection. It's about rabies. You can die from rabies, and it's a painful death. Now, the authorities could force you to get rabies shots, but they also are very painful, and they cost a lot. Rather than do that, the procedure is to hold the dog for 10 days to see whether he dies from rabies. If he doesn't, you get him back. If he dies and tests positive, you have to get the shots, which are very painful but less painful than dying from the disease.
Because rabies can be life-threatening, it is a crime to fail to quarantine your dog, or conceal him to avoid a quarantine. See for example the California law pertaining to quarantine. Furthermore, you can be sued for not quarantining the dog and thereby causing the dog bite victim to suffer from worry about rabies or having to submit to rabies injections. See Liability for failure to vaccinate.
So be glad that they have developed this method, rather than forcing you to get the shots. The bottom line: it's for your own benefit, and that of your family!