How to make a podcast from audio clips?

Another interpretation of your question just occurred to me.

Are you asking how to get them published and/or distributed?

You would have to get a host for the files, and to upload a current RSS feed page (which could be done manually in TextEdit, if you have the patience to figure out the relatively simple markup language).

Then, if you want it found by future subscribers, it's a question of getting it approved by some service like Apple's collection of feeds for the iTunes and Podcasts apps.

I'm not sure that they actually host any podcast feeds, but are probably more like a directory of feeds.

<<edit>>

Ah. Here we go: https://itunespartner.apple.com/podcasts/ - "Podcaster Support"

From your description (pasting together audio clips), it sounds like GarageBand is what you want to use.

Or Audacity (also free).