Why should you go there and how do you do it right?

I would skip this excursion. It's an elephant riding operation in the historical center of Ayutthaya, open daily from 9:00 AM to 5:00 PM. Even without visiting the ruins, you'll see the animals with their passengers on the nearby streets. Feeding, photography, and riding are the main activities here.

The reviews vary widely. Some visitors are satisfied, reporting cleanliness and that their partner enjoyed their stay. Many others, however, describe over twenty elephants chained in cramped quarters, standing in the blazing sun all day while visitors climb on and touch them. Terms like "catatonic" are used. This isn't just a single dissatisfied reviewer, but a pattern, and such impressions linger long after you've read them.

If elephant watching is important to you on this trip, there are places in Thailand that focus on observing the animals rather than riding them. That's where I would invest my money. Ayutthaya offers enough temple ruins to fill a whole day, even without an elephant ride, and Wat Mahathat with its historical park is very close by.