To summarize: Context free languages (CFLs) can be described by a context free grammar (CFG) and can be processed by a pushdown automaton.
We will he only show how to construct a PDA from a grammar - the other direction is shown in the text book (6.3.2, pp. 241).
Given a CFG , we define a PDA
Take as an example where
How does the accept ?
This example hopefully already illustrates the general idea:
The automaton we have constructed is very non-deterministic: Whenever we have a choice between different rules the automaton may silently choose one of the alternative.