R4, R5 and C3 create a clean 2.5V reference (derived off the Arduino 5V).
C1 blocks the senor's dc component.
R1 and R7 create a voltage divider which drops the peak input by 1/3.
Since the right end of R7 is tied to 2.5V, the opamp's output is centered on 2.5V.
X1 determines the gain of the OpAmp. The left end of the pot is grounded through the capacitor to minimize offset from the 2.5V reference voltage.
The lowest opamp gain that the pot will set is 1, so the overall gain ranges from 1/3 (that is why the R1/R7 divider), to a much higher value as the wiper is moved left.
The frequency response is flat from ~10Hz to an upper cutoff determined by the opamp.