A signal near 440 Hz (the center frequency) will be amplified by a factor 100 (the gain). The gain depends on all the earlier components and its purpose is to get enough signal to the LM567 in the frequency detector section. The bandwidth is 44 Hz, which is 440 Hz (the center frequency) divided by 10 (Q, the quality factor). The only frequencies appreciably amplified are those in this band, 418 - 462 Hz, in other words, 440 Hz ± 5%. Since the components that determine the center frequency are 5% tolerance, a Q of 10 is as high as it can be without risking that the filter rejects the intended frequency because the components were out of tolerance.
This design comes from the book by Horowitz and Hill titled The Art of Electronics (1989), Figure 5.19. The design is both over my head and easy to use, and makes good use of one good IC. The LM324 is easy to get and needs only a 5V supply. For the op-amps in this circuit, the reference voltage is 2.5 V. The four op-amps are interchangeable.