IC1 is an "Operational Amplifer." Essentially an op-amp is a mathematical amplifier. The purpose of all amplifiers are to take a very weak signal and make it stronger, able to "drive" more energy into something, like a speaker. When we say "impedance matching" we mean just this: using something like an amplifier to match an input signal to whatever it is being fed to.
If an op-amp's
-input were grounded, and a signal applied to the
+input, a larger but identical
+output signal would be generated, just like a traditional amplifier. So that is called a non-inverting configuration, since the same signal comes out.
Op-amps have a
-input though, which does the reverse: if the
+input was grounded, and the signal applied to
-input, the opposite signal comes out. So that would be an inverting configuration. This is very useful for many things.
In this schematic,
IC1a takes the instrument input (
+input) and outputs essentially the same thing to
Out1. That is a non-inverting configuration.
Out1 is also coupled to the
-input of IC1b, which is the inverting configuration, so the opposite signal comes out of
Out2 are differential outputs, because opposite signals are coming out of them. That's exactly what an XLR cable wants, and due to the physics behind sending signals along cables this way, it is very low-noise.
All of the other components are just for setting the voltage levels right (search for 'resistor divider'), blocking DC voltages (capacitors), and other signal-conditioning needs.