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LIME basically builds a bunch of really small linear models. It makes the assumption that although a model may be very complex overall, it can be locally be represented through linear relationships. LIME builds these linear models by picking a point and changing the data ever so slightly. Again, a good python implementation and more details can be found here.
If you are using more of a black-box method, two of the more common ways to determine how your dependent variables interact with your dependent variables are Shapley values and LIME. Shapley values are related to game theory from economics. Basically it attempts to answer how much each feature contributes to the predicted value compared to the average by looking at the average marginal contribution of a specific feature value across all potential combinations of feature values. A good python implementation and more details can be found here.