Step 1: Find your request
One thing worth noting here - some first party features and third party skills do not provide output cards for some or all of their user interactions. In cases where a card was not generated, we haven't yet found a way to retrieve the audio clips.
Step 2: Expand the voice feedback accordion
Step 3: Open Chrome's developer tools
Step 4: Go to the network tab
Step 5: Click the link
Step 6: Copy link into new tab
Alternately, you can choose to copy the link by doing a right click, and then choosing copy -> copy link address. Remember, though, that when you paste the url, it must be in a browser session where you've already logged in to alexa.amazon.com. The audio files are secured by your session data, and so using an incognito tab, or trying to hit the url after a long period of inactivity, will prompt you for a login screen.