An infinitive is, simply, a verb with a "to" preceding it. To bellow, to whine, to connive, and to go are all examples of infinitives. A split infinitive means to put some word (usually an adverb) between the "to" and the verb. To loudly bellow, to annoyingly whine, to cunningly connive, and to swiftly go are examples of split infinitives.
It's kind of an archaic rule, but editors tend to follow the older and conservative grammar rules. It really does no harm, but grammar elites might view it as ignorance.