In general it is much easier on the company if a person resigns than if they are terminated. In order to terminate a person in many states you have to document the reasons properly and fill out quite a few forms. If mistakes are made in this process then there could be consequences, most of which only arise in the case of a lawsuit or other legal claim. A termination process can drag on and be costly and a terminated employee can choose to challenge the termination in court and be awarded a substantial amount of damages as well as the potential that the court could order that the termination was voided. Employment law is complex and confusing made more complex by existing decisions. So companies often try to reduce the potential damages as well as the expense of going through the termination process. And even without a trial the person will be awarded unemployment if the conditions to deny are not met(in many states this bar is very high).
If a person resigns they have less of a legal standing should they try to file suit against the company for wrongful termination. They are most often not eligible for unemployment benefits.
So why would a person resign instead of being terminated?
It seems at first glance completely stacked against the business and making them go through the process seems the logical choice. The reason is because the terms of the resignation are right to choose that.
It could be as simple as will be offered a good recommendation. This is all that it would take if my company wanted me to resign. I know I have good skills and that I could find another job. The recommendation is worth more than the month or so salary I would get if I forced them through the process. Not to mention if they decide they no longer want me working there I do not need the stress of fighting to stay in a job where my employer no longer values my contributions.
Sometimes it comes with a severance package. In exchange for resigning a few weeks to a few months pay are offered. This pay allows you to continue to live while you find new work.
If there is reason for termination sometimes a law may have been broken. Having this disclosed may cause damage to the company, and the investigation can be costly. So often the company will offer to allow the person to resign. This has the effect of keeping the employee out of potential legal trouble and the company from having the expense and exposure.