What to do after training a classifier? In terms of application, what happens after we train a classifier? What can we learn from it? 

For example, if I trained a classifier to predict the success of a Kickstarter campaign with 80% accuracy, how can I apply this information to benefit my own Kickstarter campaign? Is there a way to know which attributes influence the outcome 'success' or 'failed' the most?
To see which attributes influence your model will depend on the type of model you are using. For Decision Trees, Random Forests, and Gradient Boosting based on trees you can plot feature importance metrics.

In linear models such as Logistic Regression, theoretically the coefficients will indicate how much the model will be influenced by their features.

Are you using Python? Let me know which models are you using so I can help you more.