Random forest output: Which features are predictive of class?

there are some general "black box" strategies for this. please see Strumbelj and Kononenko "explaining classifications for individual instances" (2008), Strumbelj et al. "explaining instance classifications with interactions of the feature values" (2009), Strumbelj and Kononenko, "An Efficient explanation of individual classifications using game theory" (2010). there is also Baehrens et al. "how to explain individual classification decisions"(2010)

There is no general-purpose way of doing that. Random forests are very nonlinear and hence use all features and some feature interactions to solve a problem. You can try model compression (Bucila et al) to get a simpler model that predicts faster and can be more easily interpreted.

If you want a clear statement of which features are most relevant it might be better to use a different, simpler model (such as the lasso) to do feature selection and then train random forests on just those features to see if the performance doesn't decrease much.

Alex's proposed strategies of model compression or using simpler models are good advice.

If you prefer to directly analyze the random forest model though, here is what you can do. Step 1: derive estimates of how strongly each input influences the model's output (maybe you have this already from the VIMP feature selection). Step 2: generate partial dependence plots to visualize the approximate relationship between important inputs and the output. I've done this and it is not too hard; you can find the write up in Mining citizen science data to predict prevalence of wild bird species to get some more details.

If you suspect statistical interactions between inputs are important (that is, the output is a non-additive function of inputs, e.g. y = x1*x2), you might want to look at Detecting Statistical Interactions with Additive Groves of Trees to check for important interactions. Then you can do partial dependence plots that show the joint effect of a pair of inputs on the output. Code implementing this algorithm is here. You should expect the additive groves to be more computationally expensive than random forests with some parameters to tune.