Why some people add results from PCA and other dimensional reductions techniques as features
This is feature engineering. You just give the algorithm another look at the data, from another point of view. It often helps to understand better data when you have different point of views.

For example, let's assume you want to learn the US road directions to a simple ML model. You give him all examples from roads 1 to 100 mapping to 0 (if the road is east-west), or 1 (road is north-south), except for the number 47. Then you ask the model to predict 47 and it will answer 0 because it's between 46 and 48 who are labelled 0.

Now if you give the model another point of view (here, the parity of the road number), then it will obviously learn efficiently the road directions.

You can view the results of the PCA as the parity, just another view of the data to understand it better.