For regression problems, a K value is first chosen, as well as a prediction point ($x_0$). The algorithm starts off by finding the K training observations that are closes to $x_0$. These training observations are denoted by $N_0$. In order to find $f(x_0)$, or the response value of your predicted point, KNN uses the average of the training observations in $N_0$. To do this, it does the following:

1. Depending on the value of k, it finds the distance (typically Euclidean) between it and the other observations.
2. It then chooses the k closest observations based on this distance ($N_0$).
3. The response values of the chosen k observations are averaged to predict the output of $x_0$.

Formally, the equation for this process is:

$\hat{f}(x_0) = \frac{1}{K} \sum_{x_i \in N_0}y_i$