The woman he calls aunt isn't really Nicholas's aunt at all. She's his cousins's aunt but still insists on being referred to as Nicholas's aunt all the same. This gives her the authority to chastise the young boy whenever he misbehaves, which is often. Nicholas' latest stunt is to claim that there's a frog in his bowl of bread and milk. His aunt won't stand for such nonsense, and so resolves to punish him. It makes no difference that there really was a frog in Nicholas' bowl—he should know, as he put it there—but as far as his aunt is concerned, Nicholas has been up to mischief again and must be punished accordingly.
In this case, that means that he will not be accompanying his cousins on their trip to the sands at Jagborough. As punishment for being such a naughty boy, he will stay at home. This method of punishment seems to be a favorite of Nicholas's aunt. Whenever the children misbehave, she has this habit of inventing some kind of festival treat of which they are to be denied. So for instance, the aunt will tell the children that she was planning on taking them to a marvelous circus in a neighboring town, but as they've behaved so badly she's not going to take them after all, and so they will stay at home as punishment.