Why do companies not want to hire over qualified people?

Generally speaking, the overqualified employee is fundamentally less likely to be happy with their position. As a consequence:

• They will likely leave at the first opportunity.
• Generally speaking, if they take the job it's because they couldn't find anything better. This can lead to a certain resentment of their situation. From the company's point of view, the best employees are the ones that feel grateful for the opportunity to work there.
• They will be more likely to get bored with the work.
• They are more likely to challenge authority.
• At some point they may change their mind about taking the pay-cut and start asking for more money.
• Some companies are suspicious of your motivations when you're aiming for a lower position. Why are you applying for this job? Are you trying to learn about the company and possibly steal trade secrets?
• (As mentioned by Ben Crowell and Vietnhi Phuvan) Many managers will feel intimidated/threatened by someone who is possibly more qualified than themselves, and will be reluctant to hire someone who might be able to take their job.

Keep in mind that hiring is a very costly and risky process for a company, especially if the new employee leaves or has to be fired, thereby starting the process all over. Costs include time spent searching through CVs, interviewing, background checks, recruitment agency fees, potential legal fees if the company has to apply for licenses or visas for the employee, overhead costs of initiating benefits, training fees, expenses for new equipment and software licenses, and possibly redundancy fees if they have to fire the employee.

That's not including the costs of potential damage to the actual work process, e.g. disruption to the projects that can occur when the new hire leaves, the opportunity cost of an unhappy employee not working very efficiently, and in the case of management positions, the damage that can arise from misleading the team and/or leaving them hanging.

So when presented with an employee that looks like a "flight risk", the company naturally starts to question whether or not it's worth the risk of them possibly leaving.

Having talked to a few Google employees, I can say that Google likes to hire on an overqualified basis. But that's because they see themselves as a growing company, and they like their employees to go at least one level up, before they reach their level of incompetence, as described by the Peter principle.

Most prospective employers' management are leery and probably insecure about offering positions to overqualified people.

1. One concern is that the overqualified individual may be taken the position to satisfy a short-term need for cash, and will bail out as soon as an opportunity that's more suitable to their quals pops up. Hiring people was fun the first time I did it but it did not take long for the fun to wear off and for me, to see hiring as a chore. Hiring people who will bail out in short order - that's not my idea of a good time to be had by all concerned.
2. Another concern may be intensified future salary/promotion demands from the overqualified individual.
3. A third concern may be the morale of the individual involved, who is taking a step down and comparing the other team members/management unfavorably to themselves.
4. A fourth concern is, why should a boss hire someone who could take their job?

Having said that, the list of possible concerns is by no means complete.

On a personal note, I've lost a bunch of interviews over the decades when as a result of my-take-no-prisoners attitude toward interviewing, I inadvertently crossed the line from convincing the boss that I could do the job to convincing the boss that I could take THEIR job.

That bosses don't like to hire overqualified people is only part of the story. I don't think that bosses like to hire people who are smarter than themselves or who can prove to be better managers than themselves either. I think the safest way is to gun for the position and not try to imply anything else.