Longitudinal microbiome data are key to understanding the dynamics of microbial communities and their relationships with the host and environment. However, analysis of such data is challenging due to high dimensionality, compositionality, irregular sampling and temporal dependencies on external covariates. Existing analytical approaches typically address only subsets of these challenges, limiting their ability to yield biologically interpretable insights. We introduce LGTM, a probabilistic modeling framework that combines flexible non-linear longitudinal modeling with interpretable topic-based representations of the microbiome. LGTM simultaneously identifies microbial co-abundance patterns ("topics") and models how their proportions change over time and in relation to host and environmental covariates. Using multiple longitudinal human gut microbiome datasets, we demonstrate that LGTM identifies diverse microbial topics whose major patterns are reproducible across runs, while achieving competitive performance in imputation and forecasting tasks. A key strength of the framework is its interpretability: LGTM yields microbial topics with biologically interpretable taxonomic compositions and directly quantifies associations between covariates and microbial dynamics. LGTM is available at https://github.com/yuanx749/lgtm.