Because when the original borders were drawn between Missouri and Illinois, the river was in a different place, and while the river will move, borders will not. The river, despite being deeply channelized and levee'd by the federal Corps of Engineers and local authorities, will change its course over time, in an event called an avulsion. The characteristic birds'-foot delta of the Mississippi and the Nile Delta are examples of channels changing over time.

There are quite a few other examples of this phenomenon: the Ste. Genevieve-Modoc ferry landing has part of Missouri in it. The town of St. Mary in Kaskasia Island is technically Illinoian, but has to be reached on land by Missouri. There's also Crains Island, Wilkinson Island, Grand Tower Island, Marquette Island, and of course the aforementioned Missouri Sister Island, where the Mississippi flows northwest and Missouri is east of Illinois. And these are just some examples from southern Illinois.

These sorts of things are more common than you'd think -- a fun example is that of the steamboat Arabia, which sank in the Missouri River in 1856 and was dug up from a cornfield in 1987-89. There's a suburb of Council Bluffs, Iowa, that's completely surrounded by Omaha, Nebraska. And so on.