This is an opportunity for you to demonstrate one of the flaws of “Sola Scriptura”. Why?
Because you’ll find that trying to nail down a definition of this theology is like trying to get a cat into a gunny sack. With the lack of central authority and a system of checks and balances (which we have in the Church), defining what is true and what is not becomes problematic.
Darby produced a translation of the Bible and initiated the concept, but C.I. Scofield was responsible for popularizing Dispensationalism. Scofield’s book really popularized it: “Rightly Dividing the Word of Truth”. After that, he produced his own Study Bible around the early 1900’s and the idea took off from there.
I think it’s safe to say that it’s about 160 years old.
As to why it hasn’t “seeped in” to the Catholic faith is because of the structure of our Church. We have checks and balances designed to make sure that such things are checked, re-checked, studied and thought about by real, academically trained theologians. Protestants don’t really have that benefit.
“Sola Scriptura” is what allows these things to “seep in” to protestant denominations. In that world, if enough people believe in a specific interpretation of scripture, then it has a very good chance of infiltrating major denominations. And, if it doesn’t, the people who believe it will can start their own denomination.
What you’ll find is that many of these advocates of Dispensationalism disagree as to the exact definition of it. Once again, these disagreements are the direct result of “Sola Scriptura”, because if you have no central authority then there is no clear definition of the doctrine.
Here’s an interview with John MacArthur that shows you an example of the disagreements.
Keep in mind that John MacArthur is a rabid anti-catholic. Use caution in suggesting your friend read that article.
You need to be careful of taking information off the web about this because opinions and definitions on this vary wildly and I find it hard to believe that you’ll find unbiased sources on the web.
Many Protestant denominations reject the concept. Most of those denominations fall under the category of “Reformed Theology”.
Here is an article that will help. Keep in mind it was written by a Reformed Theologian, so it is not a neutral position. IMO, however, it’s fairly accurate.
This is mostly from memory, so factual corrections are welcome!