Murder, She Wrote/WMG

Jessica Fletcher is a Kira
She sets up the murders and covers her tracks by covering over the Death Note and saying she's writing murder mysteries, and is able to solve the cases because she set it up that way.
 * This would explain why she solves the crimes mostly not by rational deduction, but by having randomly perceived and remembered a little detail that just happens to be the key to everything.

Jessica Fletcher always gets confessions offscreen from the killers, verbatim, before she tells everyone else.
She knew exactly what a murderer and his girlfriend had said while they were alone in a wide field.

Jessica Fletcher has ghosts help her solve crimes
Possibly those of the victim or previous murder victims.

Jessica Fletcher has the entire Northeast bugged.
Sheriff Tupper set it up, but hasn't the slightest idea what to do with it. It's all highly inconspicuous for the era.

And when she moves to NYC in a late season, she taps into their security with her new word-processing computer.

Jessica Fletcher killed all those people and pegged the blame on others to make herself famous

 * Hero System version: Here.
 * If you know Jessica, then either you or someone close to you will be killed (by Jessica) or you will be blamed for it (thanks to Jessica). The only person immune is Sheriff Tom Bosley, & even he left the series mysteriously after a few years. The only loose end is how she gets her patsies to confess.
 * She's Angela Lansbury. How do you think?
 * The German name of the series (Mord ist ihr Hobby) would translate to "murder is her hobby". Hmmm...

Jessica Fletcher is the Anthropomorphic Personification of Murder.
She naturally creates a murder-field that increases the premeditated crime rate 1000%, often retroactively, wherever she travels. Her kindly old lady facade is a ruse to lull the unwary -- Obfuscating Stupidity.

Jessica Fletcher is a Reaper with guilt issues regarding her job.
She deals with them by making sure that the Reapee's killer is caught.

Jessica Fletcher is related to Kudo Shinichi/Edogawa Conan of "Detective Conan."

 * Wait, Shinichi/Edogawa Conan killed all those people and pegged the blame on others to make himself famous?

All the murders around Jessica Fletcher are orchestrated because of Jessica Fletcher.
Someone or something is ensuring murders continually occur around Jessica Fletcher.
 * Jessica's community, neighbors, family, friends, and casual acquaintances are regularly involved in murder. Whoever or whatever is after Jessica Fletcher is intentionally and maliciously targeting everyone around her for death... and she hasn't caught on yet.
 * Going along with the Death Note theories, there's a Shinigami that has fallen in love with her. This one, however, is content to just hover invisibly without her being aware. From time to time, it snacks on the people around her. Noticing that she likes solving mysterious deaths, it arranges the murders to happen. Jessica's happy, the Shinigami gets meal service, everyone wins. Well, everyone important.
 * Let's make the best Mystery Novel Ever!!! In true Les Grossman fashion, Jessica's publisher is orchestrating everything; all to hone Jessica Fletcher's already impressive skills as a mystery author.

Cabot Cove is Evil.
Whether Jessica Fletcher realizes it or not, something about Cabot Cove compels people to murder. Cabot Cove's two percent murder rate is unnaturally high not to attract attention.
 * A Great Old One sleeps under Cabot Cove, perhaps even Cthulhu; compelling the town's residents and visitors to madness and murder. Jessica Fletcher just has more sanity points than everyone else; For now.
 * Cabot Cove is in Maine. Maine is Lovecraft Country. So it is evil, but the murder rate goes relatively unnoticed because it's no higher than anywhere else in the area. (Especially since Jessica traveled around the area!) The locals are used to it, and the people who would be shocked don't know what to look for... It took Sheriff Tupper's replacement (an ex-New Yorker) to figure out something was going on.
 * Cabot Cove is a Hellmouth, and Jessica Fletcher is its watcher. The murders are fictional accounts of vampires and demons written by Jessica (a talented mystery author) to conceal the truth.
 * Perhaps Pennywise relocated to Cabot Cove after his defeat in Derry.

Jessica Fletcher killed her husband.
Seth Hazlitt helped cover it up.

Jessica Fletcher is an earlier incarnation of Suzumiya Haruhi
She was a murder mystery writer, so obviously she subconsciously wanted to have lots of mysteries to solve. So being God/a higher stage of evolution/whatever the heck Haruhi is, she fashions the universe around herself to give her mysteries. When she finally passed away, she was reborn in Japan and continued her insanity.

Jessica Fletcher lives in a Crapsack World where people have anger management problems.
That's why so many people are killed - everyone is so Hot-Blooded. Think about it. And because of this, murder is her world's equivalent of jaywalking or theft (petty or otherwise). It's not as serious a it is here, so more people do it.

Jessica Fletcher is immortal
In the unlikely event she didn't commit the murders, she'd have to be.
 * Evidence. Though this is inconclusive because superspeed could either be her means of survival, or her M.O.
 * Perhaps, she's a Shinigami and uses her Death Note to kill people...

Jessica Fletcher will be murdered
She's such a good detective, she already knows who the murderer will be, and has already set things in place such that her murderer will confess after the wrong person is arrested at first. After it all dies down, the manuscript of her final novel, to be published posthumously, will be found, detailing all of these events in advance.

The entire series is nothing more than story ideas that Jessica has created for her books.
Think about it: She knows ALL of the victims personally...she was only the suspect in a few of the murders but that didn't last....she's able to solve all of the crimes no matter how convoluted solution...and no one seems to comment on the incongruity of an elderly woman being or near the scene of so many murders. Why does this all occur? Because all of the "murders" take place in Jessica's MIND as ideas for new novels.

Jessica Fletcher is in some way related to Richard Castle.
Both bestselling mystery novelists who help the police solve murders -- and you're telling me there's no connection?

Jessica Fletcher committed all the murders.
Chris Hardwick has suggested this, elaborating that the entire series is her hallucinations as she tries to come to terms with the horrible things she's done--Kinda like St. Elsewhere, except with psychological coping instead of autism.