Split toning is a method that is used to introduce a colour tint(s) into the highlights and/or shadows of an image.

It is used purely for aesthetic reason, and could be used to recreate the look of a particular film or just a completely unique look.

Many films have colour casts in their shadows and/or highlights. This forms part of their "look", which you may wish to recreate in a a digital image. I don't know anything about applying split toning to colour film (though I am sure there is a way), but there are several printing/toning methods that were commonly used for monochrome film, for example sepia toning and cyanotypes. This too can be emulated in Lightroom with the split toning panel.

In lightroom the toning effect is similar to that achieved by painting over with a solid colour using the adjustment brush, but the strength is dependent upon the luminosity level - i.e. it is stronger in lighter areas for the highlight tone and stronger in darker areas for the shadow tone. In lightroom, the position of the balance slider determines the cutoff between "shadows" and "highlights" for the purposes of the split toning.

One commonly used example of split-toning is to have yellow in the highlights and blue in the shadows. This yields a fairly natural looking result, as this would occur naturally when brighter areas are lit by the sun, and darker areas in shadow are lit by skylight,