Dementia (DEM) is a growing global health challenge, underscoring the need for early and accurate diagnosis. Electroencephalography (EEG) provides a non-invasive window into brain activity, but conventional methods struggle to capture its transient complexity. We present the \textbf{EEG Microstate Analysis Framework (EEG-MSAF)}, an end-to-end pipeline that leverages EEG microstates discrete, quasi-stable topographies to identify DEM-related biomarkers and distinguish DEM, mild cognitive impairment (MCI), and normal cognition (NC). EEG-MSAF comprises three stages: (1) automated microstate feature extraction, (2) classification with machine learning (ML), and (3) feature ranking using Shapley Additive Explanations (SHAP) to highlight key biomarkers. We evaluate on two EEG datasets: the public Chung-Ang University EEG (CAUEEG) dataset and a clinical cohort from Thessaloniki Hospital. Our framework demonstrates strong performance and generalizability. On CAUEEG, EEG-MSAF-SVM achieves \textbf{89\% $\pm$ 0.01 accuracy}, surpassing the deep learning baseline CEEDNET by \textbf{19.3\%}. On the Thessaloniki dataset, it reaches \textbf{95\% $\pm$ 0.01 accuracy}, comparable to EEGConvNeXt. SHAP analysis identifies mean correlation and occurrence as the most informative metrics: disruption of microstate C (salience/attention network) dominates DEM prediction, while microstate F, a novel default-mode pattern, emerges as a key early biomarker for both MCI and DEM. By combining accuracy, generalizability, and interpretability, EEG-MSAF advances EEG-based dementia diagnosis and sheds light on brain dynamics across the cognitive spectrum.