Reward boards are the main collectible in the game, and this achievement is earned by breaking all of them. There are 50 Travel Discount signs (yellow) and 100 Credit Reward signs (green), which can be found in 1000, 5000, and 10,000 Cr flavors (the bigger the Cr dot on the sign, the bigger the reward). You can try and find them on your own, so if you're one of those people, have fun with that. You can use a video guide online and save yourself the 300k Cr for the Treasure Map, so if you're thrifty with in-game currency, I've included the YouTube link for that (Credit to VGFAQ for that). Or, as I did, you can buy the Treasure Map from the World Map screen in the pause menu. Go into the map and press RB if you have the 300,000 Cr for it, and it'll light up every reward sign on the map (and in your mini map as you're driving). My advice?
BUY THE TREASURE MAP.
You make way more than enough money in the game via the grind it'll take to master every event (and to finish the online side of things, as well), and there's really no specific reason to be frugal with your Credits really at any point after you've beaten the Horizon Finale. Buy the map and make it easier on yourself to get the 150 boards, because it's a big freaking map to hide stuff on, and they do a good job hiding more than a few of these.
After you've bought the Treasure Map, go into your world map and press LB to access the map filter. Turn everything off, and then just turn on the reward signs, and then go about your day of driving to all the little green and white (Travel Discount boards are white on the World Map) rectangles. Once you've smashed a sign, it turns a translucent grey on your map and you can forget that it ever existed.
Your achievement will pop after you smash your 150th reward sign.