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After you've obtained and repaired the first water pump, Casey will ask you to go to the Church yard and install it at a blinking red pump station. At this point the game will introduce you to guts mode through a small tutorial.

Any walker that's 'dripping' drops guts once you kill it and cut open its belly.
Take the guts and make a rubbing motion back and forth across your chest until the circle icon is full, then your screen will turn red and you'll see the eyeball icon on your hud like a buff-timer. If you sprint the buff falls off.

In my game, the next time I returned to the church yard I noticed a bunch of guts laying around where the first pump station was, probably because that game despawned all the guts-walkers that were meant for the tutorial. I found at least five.

I don't believe you can reload save to accumulate progress however once you've used guts you can sprint out of it and it will still count, no need to wait for whole duration.