I was thinking about this the last time we went en famille on a big trip. It was at Easter, and we were flying to England for a week. When we arrived at Logan Airport in Boston and were checking in, I suddenly remembered that I had recently joined British Airways’ frequent flyer programme. I also remembered that I had put the card in the carry-on bag that was hanging around my neck. And here’s where the trouble started.
In the second paragraph, the narrator starts describing an incident from the narrator's life. The incident took place during the previous travel he took with his family. It was during Easter, and they were flying to England from Boston, U.S.A. It was a big trip, and they were planning to spend a week there.

The family arrived at the Logan Airport in Boston. While they were in the line to check-in, the narrator suddenly remembered the privilege card he had with him. He had recently joined British Airways’ frequent flyer programme, and by using the card, he would be eligible to gain more points.

The programme works like this: the more the members travel, the more points they gain. With the points they have earned, they will be eligible for a number of privileges. For instance, they will be able to redeem the points for a free flight or become eligible for a free (or with a discount) holiday package. However, the member should produce the card at the desk before checking in. If not, the journey will not be considered and, as a result, it will not yield any points to the member.

Hence, the narrator starts searching for the card. He remembered putting the card in his carry-on bag which was hanging around his neck.

The paragraph ends with the narrator saying that all the trouble started there. The concluding statement of the paragraph suggests an impending humourous situation.
A check-in line in an airport
Meanings of difficult words from the paragraph:
En familleWith one's family
National Council of Educational Research and Training (2006). Moments. Accidental Tourist- Bill Bryson (pp. 56-59). Published at the Publication Division by the Secretary, National Council of Educational Research and Training, Sri Aurobindo Marg, New Delhi.