What is the need for the extra resistors in this equalizer circuit?
Yes they are part of the filter. The top section relies on R4 and R5 and C1 for shaping the frequency response of the op-amp circuit. The 100k pot has no part to play in the filtering other than guiding the current that reaches the inverting input of the op-amp from either the actual input or the op-amp's output.

Maybe try researching Baxandall tone controls: