Logistic regression if 3 categories in outcome variable
Logistic regression can work on multi-class. Frankly it is not big different from binary classifier. The question is how does it find the coefficients for each class which you had displayed. By default to find the coefficients for a single class, it goes with one vs rest combination. So setosa vs remaining all classes. It gets coefficients iteratively by minimizing cross entropy. So ultimately you will end up with a matrix having a size of n_classes vs features.

Coming to the interpretation part of the question. To interpret the importance of a single feature, you can do it as - Given the coefficients of every other feature to be same (in comparing a class with another) including intercept, the class that has the feature with highest coefficient has greater chance of engulfing the new point. This being said the features having high petal length and width and low sepal length and width contribute good for a data point to belong to class virginica. But it does not make much sense to interpret each coefficient individually. Hope I made some sense.