How does an individual feature affect predictions in neural network classification problem?
Is there a way to quantify the likelihood in ANN similar to logistic regression wherein regression coefficients give the change in the log odds of the outcome for a one unit increase in the predictor variable?


Good question. 

Yes, there is a way. The approach that can help you is called partial dependence plot (PDP), see the links below for further details and examples. 

The approach is model agnostic, i.e. works well for any predictive model, powerful yet simple. 

The main steps for one-dimensional partial dependence plot are as follows


Fit your model as usual
Select the predictor of interest and a set of values to be investigated (e.g. income as in the article you refer to and values of say 50k, 70k, 80k, ..., 120k)
For all observations in your dataset replace the values of your predictor with the first value from the set above (50k). 
Calculate the model output for the modified dataset from the previous step and calculate the average over all observations. 
Repeat steps 3-4 for the remaining values (70k, 80k, ...) and plot the values of your predictor along X axis and the corresponding averaged model predictions along Y axis. 


With one-dimensional PDP described above you can easily see the marginal impact of a predictor being analysed on the model output. Furthermore, one can use similar technique to perform multi-dimensional analysis, e.g. to investigate the impact of interactions. 

partial dependence plots- scikit-learn documentation

partial dependence plot - tutorial by Dans Becker on Kaggle