Once upon a time, back in the 17th or 18th century, well before I was even in grade school, English grammarians had high hopes for their language. They wanted to polish it like marble and make it follow the rules of Latin grammar. For one thing, they tried to stamp out the practice of ending sentences with prepositions - a fussiness that Winston Churchill famously dismissed (according to legend, anyway) as nonsense up with which he would not put.
And they wanted to mend split infinitives, to banish any interloping modifier intruding upon the sweet unity of a verb with its preceding "to." They would have cringed at one of the most famous split infinitives of contemporary pop culture, the Star Trek motto "to boldly go where no man has gone before."
That "boldly" is misplaced! I can imagine a gaggle of grammarians gasping in horror. "The 'to' and the 'go' should adhere like glue! Away with the 'boldly'! It belongs after 'go'!"
Now just what is an infinitive? You can think of it as a verb form that is keeping its options open. It "expresses existence or action without reference to person, number, or tense and can be used as a noun," as one dictionary puts it. It hasn't been "limited," or made "finite" by being broken out of its shrink wrap and plugged into a sentence as a predicate verb. Compare the airy abstraction, the generic sweep, of "To drive into town during rush hour would be foolish" with the more concrete, "He drove into town at 8:30 this morning."
Most of us get this right most of the time. But controversy over the split infinitive ("to boldly go") lives on. People really do bring split infinitives up with copy editors in social situations where a bit of small talk is called for. "Hmm, by the way, how do you feel about split infinitives?"