What is the purpose of a Victor Airway?
An airway (Victor or otherwise) is just a standard route for aircraft to fly on in the national airspace system - for all practical purposes they are literally highways in the sky (with the lanes being stacked vertically rather than horizontally).  

This is part of the paradoxical logic of how ATC provides separation services for IFR aircraft: "You have to put them together to keep them apart." -- In an environment with no radar, or in crowded airspace, having everyone following a standard routing ensures that you know what each plane is going to do. If you have 10 aircraft all flying along an airway and each separated by altitude or time you know they won't hit each other

We define the attributes (width, minimum altitude, etc.) of a victor airway mainly for purposes of obstacle clearance:  

A VOR has a specific degree of precision in its measurement, and if we assume the maximum distance from the VOR (the widest degree of arc) and the maximum allowable "slop" in the VOR receiver/indicator in the aircraft that means an aircraft that thinks it's "on the airway" will be somewhere within a given width around the defining radial: For Victor airways that width is 8 nautical miles - 4NM on either side of the centerline radial - or 4.5 degrees if the VORs are more than 102 nautical miles apart.
Those numbers may sound  like they're arbitrary, but they aren't: If the VORs are exactly 102 nautical miles apart then the width of the airway at the 51 nautical mile midpoint between the two VORs is - you guessed it - 4 nautical miles. The further apart the VORs are the wider the center of the corridor becomes, accounting for the fact that a degree of arc gets wider as you get further from the VOR.

Airways are an IFR construct so if we continue on the assumption that the plane is in the clouds and the pilot can't see anything then within the airway corridor the aircraft needs to be clear of any obstacles (towers, trees, etc.), and of course it needs to be able to receive the VORs that define the airway. The combination of those factors and a few others will determine the minimum altitude at which a given airway can be used.



Note that much like ground highways airways are not always the most direct or convenient way to fly from one location to another. In particular the Victor airways are defined by VOR radials (or the intersections of such radials), but they're not always a straight line between two VORs.
As an example, take a look at this utterly contrived flight plan for the purposes of illustration: The airway I picked (V451) will take me from Groton VOR (GON) to La Guardia VOR (LGA), but it bends in three places on this route: CREAM, KEYED, and NESSI (not labeled on the sectional, but its the bend southeast of Bridgeport). A straight line between the two VORs would be the 265 radial from Groton direct to La Guardia: A much shorter distance, but not a "standard" airway routing.