"Ich ließ den Hund im Garten."
I would think that "I let the dog in the garden" would mean that you opened the garden gate and allowed the dog to go inside. "I left the dog in the garden," to me, would mean that you were leaving to go somewhere else and you put your dog into the garden while you left. You might still be with the dog in the first instance, but you're certainly not in the second. Does that help?