Why are people subject to arrest when stabbing someone in self defense?

Even if there was video or witness accounts that it was self defense, the police are there to bring peace. In bringing peace they usually will arrest everyone involved in the altercation. I was told once even by a police officer I called over someone threatening to come shoot my house up that if the person did show up that I had every right to defend my home and safety with my own gun. However, I would likely go to jail if I shot the guy, certainly would go to jail if I killed him. The police aren't the court system. They aren't there to prove or disprove anything. Probable cause is what police work on and all they need to arrest. Whether it was self defense or not ultimately rests in the hands of the district attorney and the judge relying upon factual evidence presented, if they choose to prosecute. Even if you had for instance a concealed hand gun license, or license to carry, and you shot and killed someone in your home or on your property while they were attacking you, you will likely still go to jail at least until they determine or they are convinced it was more probable than not that you acted in self defense. They can very well lock you up and let the court decide.

People who intentionally murder other people try to claim self-defense all the time.  What would prevent someone from murdering another person and then using self-defense as an excuse?  People would be getting away with murder all the time.

The fact is you stabbed someone now someone else has to decide whether it really was self-defense or murder.  That's just the way the system works.

Because self defense is an "affirmative defense" which means the burden is on you to prove you had the privilege to use it. 

Also, many people don't understand self defense laws, so it is not unusual for someone to use excessive or unreasonable force. A common example, "he hit me first" is NOT a valid argument for self defense. 

EVERYONE gets arrested while the Police figure it out.  Now, why did you carry a knife?

you get to make your case to a jury

In a case like that - it could take YEARS to prove self-defense REGARDLESS of what weapon you use!

And in most cases, it does at least initially sound suspicious - first, no it's not unheard of, but not real common to get attacked like that - AND when it does, you just happened to have a defensive weapon on you at the time! - It's not as though most people carry weapons all the time any more!

Also, even if the use was justifiable, the weapon itself may not be legal to carry, like in the case of a knife, if the blade is too long...

There must have been more to the story than that. Was the victim stabbed repeatedly? Did the person who was arrested initiate the situation? Did they strike first? These are all questions the police would be asking during their investigation. Also, check the self defense laws in your state. Many states have different self defense laws.

Because you stabbed someone. Yes, the burden of proof falls on the police & prosecution to prove that your actions were criminal, not the other way around, but even so, of course you'd get arrested. Although, if the police feel like there's no proof of criminal intent they'll ultimately let you go. They'd only charge you if they feel that your stabbing of the other person was committed under criminal responsibility. They have to investigate these things though, naturally. They can't and shouldn't just take your word at total face value; although obviously the benefit of the doubt is always given to the victim.