A violin plot is just a histogram (or more often a smoothed variant like a kernel density) turned on its side and mirrored. Any textbook that teaches you how to interpret histograms should give you the intuition you seek. Edit per Nick Cox's suggestion: Freedman, Pisani, Purves, Statistics covers histograms.
As far as interpreting them in a more formal way, the whole point of graphing the distribution is to see things that statistical tests might be fooled by.
One thing I like to do with violin plots is add lines for the median, mean, etc. Sometimes I'll superimpose a boxplot so I can see even more in the way of summary statistics.
At very least, you should be able to pick out any gross deviations in the first few moments (mean, dispersion, skewness, kurtosis) as well as bimodality and outliers.