For such a prominent part of the geography of the US, there's a frustrating lack of historic resources regarding section-line roads. The section lines came first, as that's how the land was surveyed and sold. Beginning with the old Northwest Territory, nearly all of the US west of the Appalachians was surveyed into six-mile by six-mile townships, further subdivided into 1-mile squares known as sections. In farm country, a full square mile (640 acres) was more than most farmers could afford or work, so farms were commonly 80- or 160-acre partial sections. (Our expression "back forty" comes from this common farming pattern.)

In general, most states in the 1860s passed laws enabling counties to declare that a right of way 66 feet wide (33 feet taken from the adjacent land on both sides) along all section lines was a public road.

As farmland later became city subdivisions, the farmsteads were usually subdivided evenly until an optimal block size (often 1/16 mile x 1/8 mile) was reached. Here's a map showing this process on Chicago's Northwest Side.

Andros Linklater's book Measuring America is a fascinating look at the rectangular survey system and related topics. The only book I have that discusses section-line roads is Hildegard Binder Johnson's Order Upon the Land, which focuses mostly on Minnesota.

The concept of street grids, or grid plans, are at least 4-5000 years old, with cities as far back as the Indus Valley Civilization in the Bronze age planning their cities using grids.

As far as I know, it was done mainly for simplicity. First off, it'd be much easier to specify a location if you had a set system for streets, secondly, it would be much easier to divide land for people to buy or rent.

PS. High school history teacher, bit rusty on this subject.