What is the purpose of the recurrence relation in low and high pass (audio) filters?
An IIR filter is a recursive one. It means, current value of output $y[n]$ is computed based on its previous value(s).
Consider a simple 1st order IIR filter definition:
$$y[n] = a \cdot y[n-1]+ b \cdot x[n]$$
$a$ and $b$ are the filter coefficients.
Now assume you have a long block of N samples input data. If you want to process that block of input data with this filter, you can do it like in this matlab code :
N = 1024;       % number of input samples
x = randn(N,1); % some random input.
a = 0.5;        % IIR filter coefficients
b = 0.9;

yi = 0;         % initial value for output
y = zeros(N,1); % matrix (array) to hold processed output samples

y(1) = a*yi + b*x(1)   % assign first output sample based on y(-1) 
for n=2:N              % assign rest
    y(n) = a*y(n-1) + b*x(n);
end