In short, the name France comes from a Germanic tribe the eventually ruled much of Central Europe: the Franks. Their realm fragmented over time, but they continually held the lands that would eventually become the country we know as France. This control lead the expansion of the name of the people to also refer to the land.
What would become the word 'Franks' first comes up in the 3rd Century in Roman writings. They refer to a group of tribes that shared a language and culture, but were distinct entities in themselves. They lived on the eastern banks of the Rhine River, which Rome had established as its frontier. In the 4th Century, they crossed the river into Gaul as the power of the Romans diminished. Over time various individuals and groups of Franks became allies of Rome and even were part of its military organization.
In the later half of the 5th Century, King Clovis I of the Salian Franks expanded his realm to encompass much of what is now France, Belgium, and parts of the Germany. In the wake of the collapse of the Western Roman authority, many Germanic tribes and kings were making claims for power.
The name 'France' comes from the term Francia, which denoted the kingdoms and lands of the Frankish peoples around the time of the fall of the Western Roman Empire. Initially the Franks were a small Germanic kingdom along what is now the French-Belgian border from the Strait of Dover to around Maubeuge, France. Under Clovis, the kingdom expanded to include much of present-day France (minus Brittany and the southeast corner), Belgium, as well as parts of Germany and the Netherlands.
The realm fractured and grew repeatedly over time until Charlemagne expanded the kingdom to encompass continental Europe from the Pyrennes to a line from Denmark to the Adriatic sea, including much of northern Italy. With his passing, the kingdom was split into three parts for each of his sons: West, Middle, and East Francia. While the Middle and East kingdoms fragmented, the Western one remained a single unit. Its rulers called themselves Kings of the Franks until Phillip II in 1190, who referred to himself as the King of France.
The 'k' sound in 'Franks' turned into a 'ch' sound in Ecclesiastical Latin. In the local variations of Latin that evolved into Old French, it became a 'ts' sound, and later dropping the 't' to just become 's'.