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Choose $\lambda$ to minimize an estimate of out-of-sample mean squared prediction error, commonly K-fold cross-validation or a separate validation set. As $\lambda$ increases, the coefficient vector is shrunk toward zero. This generally increases bias of an estimator but decreases variance of an estimator; the minimizing value balances the two contributions in the bias-variance tradeoff. The independent test set in the question can assess the final choice, but repeatedly selecting $\lambda$ on that same set would cause data leakage.

The $L$-nearest-neighbour classifier finds the $L$ training predictors nearest to $x$ and returns the majority class among their labels. Increasing $L$ averages more labels and reduces variance of an estimator, but uses observations farther from $x$ and therefore increases bias of an estimator; decreasing $L$ reverses this bias-variance tradeoff.