What does the radio will never take the place of the newspaper because you can't swat flies with it mean?

It's a rather humorous comment on the progress of technology. When radio came on the scene, people feared the demise of newspapers. The quote is saying that people have gotten used to the alternate uses (line bottom of birdcages, wrap garbage, kindling for fires, swat flies, etc.) which could save newspapers from oblivion.

Perhaps a more updated version could be: "Kindles will never take the place of books because you can't stand on them to reach a higher shelf."

(from a previous answer: swat means hit somebody or something with a sharp blow or slap.)