How does density altitude affect performance, temperature, and pressure; and how is it calculated?
The International Standard Atmosphere (ISA) gives “average” values of air pressure and density by altitude. It starts by defining standard pressure 1013.25 hPa (29.92 inHg) and standard temperature 15°C at sea level, and derives the values for higher altitude from physical laws.

A “density altitude” is altitude in ISA at which air has the density you are observing. This is analogous to “pressure altitude”, which is altitude in ISA where the pressure you are observing occurs.

You can measure pressure altitude easily by setting your altimeter to the standard pressure of 29.92 inHg/1013 hPa. However you don't have any instrument to directly measure air density, so you have to calculate density altitude from pressure altitude, temperature and humidity.


  And why 8300 ft specifically? Was it calculated? If so, how?


This AOPA document provides approximate formula:


  Density altitude in feet = pressure altitude in feet + (120 x (OAT - ISA temperature))
  
  
  Pressure altitude is determined by setting the altimeter to 29.92 and reading the altitude indicated on the altimeter.
  OAT stands for outside air temperature (in degrees Celsius).
  ISA stands for standard temperature (in degrees Celsius).
  
  
  Keep in mind the standard temperature is 15 degrees C but only at sea level. It decreases about 2 degrees C (or 3.5 degrees F) per 1,000 feet of altitude above sea level.


I have also found a calculator, which uses more accurate equation, also taking humidity into account. It comes with detailed explanation of the physics involved.

Giving it current METAR FAGM 301600Z 33006KT CAVOK 27/00 Q1022 it calculates density 0.967 kg/m³, which is expressed as density altitude 7 866 ft (2 397 m). 

It's already late afternoon, so earlier in the day it might have even 8 300 ft (if it was 31°C, which might have been just past noon)


  What does the 8300 ft mean in terms of air quality?


It is simply another way of expressing the density 0.954 kg/m³. You are normally familiar with how the performance decreases when you climb, so expressing density as altitude allows you to easily reference the loss of performance due to the hot weather to that.


  What does the 8300 ft mean? Does it mean that at that altitude, aircraft performance will be at its lowest?


It means aircraft performance will be as if you were at 8 300 ft on a standard day.

Both aerodynamic and engine performance depend on density. For aerodynamic performance higher density altitude means you'll need higher true airspeed for the same indicated airspeed, and for engine higher density altitude (lower density) means lower maximum power.

For an average single-engine piston aircraft, 8 300 ft is quite high. The performance will be poor.