The thing about LEDs in general is that we need to pull them down because, in the old days, TTL could pull down a lot more than it could pull up.
Given that we pull down to turn on, it is kind of backwards to how we view logic. In fact, there is an entire way of thinking about negative true logic.
With the inverter things are easy to understand: Logic '1' to turn the LED on, logic '0' to turn the LED off. Wired the same way to an Arduino output (pull down configuration), the logic is upside down. Logic '0' to turn the LED on, logic '1' to turn the LED off.
It all works out...