Let's think of an example which would better help visualize this situation.
1) Someone exercises a lot so they can concentrate more on their homework (here running and concentration on homework are dependent events)
2) Someone exercises a lot because they wear a blue t-shirt (here, you can clearly see that wearing a blue shirt and exercising are not related. These events are independent).
Now concentrating on P(a | b) = Probability a occurred if b occurred.
It does not matter if b occurred (just like it didn't matter that the person wore a blue shirt which meant that they exercised) for the outcome a to occur.
