Understanding additive function approximation or Understanding matching pursuit

I am trying to read Greedy function approximation: A gradient boosting machine.

On page 4 (it is marked as page 1192) under 3. Finite data the author tells how the function approximation approach breaks down when we have finite data and some way to impose smoothness is needed to get a function that can be used at points other than the ones provided in the training dataset. One way it suggests is to use parametric base functions (like in neural networks) and in case that is infeasible, we can try a 'greedy stagewise' approach. This is called 'matching pursuit' in signal processing.

I understand how, in a non-parametric approach, you cannot make a prediction at points other than the ones provided in the training dataset. I understand why a parametric approach would work to solve that issue. I don't understand :

What are the implications and significance of 'greedy' and 'stagewise' in the 'greedy stagewise' approach? Can you please explain what is being said in that paragraph.