A question after my own heart!
There are a few difficulties with responding to such an enormous question. The first is that there are 22 Eastern and Oriental Catholic Churches and every one of them has its unique history, culture, tradition, liturgy, and so on.
The second is that most of the documents written specifically about those churches were written within their native tongues.
The third is that every person who reads or speaks such a large and vague question really has a different set of particular questions which they interpret it to mean. One might want to know the history. One might question if they’re Catholic. One might want to pursue some of the spirituality. One might be looking for a Novus Ordo alternative. Every person who responds similarly has personal interpretations for what it means that shapes how he responds.
I would propose that focusing on one Eastern Catholic Church would be a good start. The Ukrainian Greek-Catholic Church is probably a good choice for several reasons. They’re the largest Eastern Catholic Church, they have many English resources in comparison to other Eastern Catholic Churches, they don’t have any significant controversies which will cause someone who doesn’t know anything about the Eastern Catholic Churches to be overwhelmed up front, and there’s a lot of documentation about them from numerous points of view.
UGCC official website. This website covers the history well.
St. Elias UGCC in Brampton, Ontario, Canada. This website has a lot of cultural content like definitions, church ettiquette, and videos.
EWTN. This webpage has an English translation of the Union of Brest which reunited the Ukrainians and the Roman Catholics.
Once you’ve read that, which isn’t much, then you’ll probably have some more specific questions that you could then ask.