IMPORTANT: You will not find these drops on the easiest (Drizzle) difficulty, so you must play on either the Rainstorm or Monsoon difficulty for them to ever show up (I found mine all on Rainstorm). Once you pick up a specific monster's log, that log won't ever show up again.
Monster Logs are random drops from enemies that you've killed, and the icon is a slanted book that simply says "LOG". The achievement will pop once you grab your 15th one. Keep in mind, there are 31 different monster types in the game, so it can be any 15 of those.
You can check to see which specific logs you already have from the main menu. I found it helpful to know which monster logs I already had, so I would stop focusing on killing those enemies if I didn't absolutely need to. If I knew I needed a specific monster, I would hang out in the biome where that monster tended to show up and grind out kills until I got the log I needed.
As this is obviously luck-based, this may very well be your last achievement in the game (I got it right before getting my 50th death achievement). The deeper you get into a run, the more logs you'll likely collect, as you'll encounter a wider variety of enemies. My first time beating the game on the Rainstorm difficulty, I played for well over an hour, and picked up 9 logs.
For anyone wondering, here's a list of all the monsters, per the Risk of Rain wiki: