"Up with chaos! Down at grammar!"
What's the big deal? Well, originally "hopefully" was supposed to be just a regular old adverb, meaning "in a hopeful manner." Here's an example:
"The tedious pedant corrected Internet grammar mistakes hopefully, mindless of the fact that everyone hated him."
Using this definition of the word "hopefully" to start a sentence doesn't make any sense. But it turns out that "hopefully" has another definition, where it serves as a sentence adverb. In that role, it's able to modify an entire sentence, instead of just a crummy little verb. This just happens to be the definition of "hopefully" that most people use. It's definitely a younger definition, having only shown up sometime in the 20th century, but it's been widely used since then and roughly parallels other widely accepted sentence adverbs, like "frankly" or "mercifully or "dongtacularly."