Why does the number of lines per unit area through a perpendicular area indicative of the strength/magnitude of an electric field? Is there an intuitive reason for why this makes sense? I thought that ultimately a test charge experiences a different force depending on its exact position in the field why would the nearby lines matter?
 A: Don't be under the impression that Electric field is present only on the lines and is zero in between. Field Lines are just a way of representing vector fields. Its "a visual aid".
It is a convention that we represent higher strength field with more number of line. You could draw it the other way round as well.
A: 
Why does the number of lines per unit area through a perpendicular
area indicative of the strength/magnitude of an electric field?

They aren't. They can convey information on the relative strength of the field at different locations of the same diagram, based on the relative densities of the lines, but no quantitative information on the actual field strength.
The number of field lines on a diagram has no quantitative meaning. Electric field lines are simply a useful tool to provide primarily qualitative information on the nature of the electric field. This information consists primarily of:

*

*Arrows that show the direction of the field at a location. By convention, the direction of the field is the direction of the force that a positive charge would experience if placed in the field. Consequently, it provides information on the nature of the source of the field (positive or negative charge). But it does not provide quantitative information on the amount of charge.


*Line density (lines per unit area) providing information on the relative strength of the field at a location compared to another location on the same diagram.  But it does not provide quantitative information on the actual electric field strength.


*Application of Gauss's law and a Gaussian surface. The field lines may provide information on whether the net charge at within a specific area (or volume) is positive, negative, or zero. But it does not provide information on the actual amount of negative or positive charge.
Hope this helps.
A: The electric field lines are simply defined that way. Their definition gives us a proper visual aid to get a rough idea about how a field behaves in a region.
You can argue that the Gauss law given by: $$\Phi = \oint{\textbf{E}\cdot d\textbf{A}}= \frac{q_{enc}}{\epsilon_o}$$ Gives a direct correlation between the flux(the amount of field lines passing through a surface) and the field at that region but that is hardly intuitive anyway.
