The speaker, "Beth," is trying to write in a colorful and entertaining way. It is informal. It is storytelling.
She is describing a long period of time. She is describing it in two different ways at the same time. She is saying that the time period was twenty-two years, and during that time the Zapps had three children.
In ordinary life, as we experience it, we are not precise timekeepers.
She could say "twenty-two years later." That's a number. It's a statistic. We can also measure life by means of life events. "Three children later" might be the way we feel the passage of time if we are parents. Putting both ideas together in a short sentence, she says "twenty-two years and three children later..."
The sentence actually carries the idea a step further. The same period of time includes:
a) Twenty-two years
b) The births of three children
c) Visits to over a hundred countries
It sounds like a rich, busy life.
She could have even written "Twenty two years, three children, and a hundred countries later..."
Here's another example of how you could use that pattern. Let's say I have always liked Toyota cars.
"I bought my first Toyota in 2000. I loved it, and my next three cars have all been Toyotas"
Or, we could measure the time period in cars instead of years:
"I bought my first Toyota in 2000, and three Toyotas later I still love Toyotas."
Or, we could combine both ideas into one sentence:
"I bought my first Toyota in 2000. Twenty-two years and three cars later, I still love Toyotas."