Earth observation involves collecting, analyzing, and processing an ever-growing mass of data. Automatically harvesting information is crucial for addressing significant societal, economic, and environmental challenges, ranging from environmental monitoring to urban planning and disaster management. However, the high dimensionality of these data poses challenges in terms of sparsity, inefficiency, and the curse of dimensionality, which limits the effectiveness of machine learning models. Dimensionality reduction (DR) techniques, specifically feature extraction, address these challenges by preserving essential data properties while reducing complexity and enhancing tasks such as data compression, cleaning, fusion, visualization, anomaly detection, and prediction. This review provides a handbook for leveraging DR across the RS data value chain and identifies opportunities for under-explored DR algorithms and their application in future research.