All in-app purchases in Little Big Robots are processed by Apple through the Apple AppStore.
As such, you can view detailed information about purchases made through the game in the purchase history of your AppleID.
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On your iPhone, iPad, or iPod touch:
Settings → Your account name (Apple ID, iCloud, iTunes & App Store) → Media & Purchases → Purchase History -
In iTunes (PC/Mac):
iTunes → Account → View My Account → Sign in with your Apple ID → Purchase history → View all -
On the Web:
Go to reportaproblem.apple.com → Sign in with your Apple ID → expand order details.
Read more about how to view your iTunes & AppStore purchase history here.