To answer the question of how Phineas Gage' brain accident helped Doctors better understand the human brain and in a more general sense the human mind, first we need to know some details about the case.
In September 1848 the life of a young worker of the train rails turned around forever due to a working accident. Phineas' work consisted on blowing up rocks using explosives using black powder. It was because of that that the accident took place.
The accident of Phineas is important because it changed the way doctors, more specifically neurologists; see the human brain and their functions. Before the accident and the later documentation of it, doctors had never associated the physical aspects of the brain to behavior, emotions and the "soul". They had never linked that the emotions and actions could be affected by specific parts of the brain.
Here we have evidence to support these assumptions:
"The damage to Gage’s frontal cortex caused by the iron rod seems to have resulted in a loss of social inhibitions. The role of the frontal cortex in social cognition and decision making is now well-recognized; in the 19th century, however, neurologists were only just beginning to realize these connections. Gage’s injuries provided some of the first evidence that the frontal cortex was involved in personality and behavior."
As we can read in the citation above, it was due to Gage's accident that the doctors started to link to a specific area of the brain, the Frontal Cortex, to behavioral changes.