Augmenter tubes augment airflow through the nacelle; they use exhaust gas passing through the augmenter tube as a jet pump to draw more cooling air through the nacelle, and specifically to direct that cooling air for more accurate, efficient intracowl airflow (cooling airflow inside the cowling).
Exhaust gasses are directed through the augmenter tube. and both the exhaust gas flow and the positioning of the tube itself draw air from inside the engine nacelle and cowl area into the augmenter tube to increase, direct, and assist airflow.