I'm also curious about this with Thai food and peppers. Was thai food always spicy and they just used black peppers up until the introduction of new world peppers, or is Thai food being spicy a modern thing?
I asked a similar question a little while back and got a great answer! Only regarding Tomatoes, not Peppers, so I wouldn't want to discourage other answers here, but its a start, and I know /u/caffarelli is a mod here so might be around to answer follow ups on it anyways.
Since the history of Tomatoes and peppers are different, let me start with tomatoes, echoing a bit of what was said in this post that has already been mentioned in the comments.
The first thing you have to do when talking about tomato history is get the modern day version of the fruit out of your head. The tomatoes of the past were unreliable crops, and had a very quick "shelf life". They were also quite bitter, leading many people to believe they were poisonous. They simply weren't all that popular in the New World. But they looked interesting, which helped popularize them to some extent when they exported from the Americas to Europe, and the Mediterranean climate more explicitly.
Because they were deemed "inappropriate to eat" by those wealthy enough to afford gardens, the tomato as food item became popularized (to some extent, limited by their shortened from-ripened-to-rotten timeframe) by the poor. This was helped by the tomatoes low-maintenance growth ability. As gardeners will tell you, it's difficult to screw up growing a tomato, especially in the Mediterranean type climates.
Tomatoes popularity took off with the Industrial revolution, specifically when canning became popular, both on a home scale AND an industrial one. And because of the higher acidity in Tomatoes, they didn't lose as much flavor in the canning process as other vegetables, as the acid works as a preservative. Compare that against the flavor of canned peas or canned potatoes, and the it's easy to see why the Tomato was preferred. Once the market showed a preference, several companies came in to supply the burgeoning demand.
To put it another way, tomatoes grow like weeds in Italy, which makes them prevalent. Because they were prevalent, they were cheap, which made the masses happy. And because of their acid content, they survive canning quite well, making even more popular than other veggies.
This is sort of the high level overview, and definitely pare it with the link noted above and elsewhere. I also recommend Food in Early Modern Europe by Ken Albala and Italian Cuisine: A Cultural History by Alberto Capatti & Massimo Montanari to fill in the many, many details I've overlooked or glossed over. I'll try to address the Pepper side of the equation in a separate response.
You may also find an answer at /r/AskCulinary.
Okay, so Pepper has a far more difficult history to understand then that of tomatoes. The problem is that...well look at it this way - We have green, yellow, orange, and sometimes even purple bell peppers out there in the grocery store. They all look different in colors (let's ignore the shape, because today's farming considers excessive shape variety a bug rather than a feature). But we know they are all still bell peppers.
Back in the day of the Portuguese (and then later Dutch, French, Spanish, and English) explorations into South America, different botanists would see different variations of the same genus, but document them as divergent. You can read about this difficulty in taxonomy at this link from Purdue.
The pepper we're concerned about is called Capsicum annuum, and it runs the spectrum from Bell Peppers to jalepeno and habanero peppers. And while you mention it's popularity in Italy, it's not really as prevalent as you might think. The primary drivers of its popularity were the Portuguese, and the prevalence of peppers in the cuisines of Africa and South Asia (specifically India) follow the Portuguese trade routes. The first documentation of peppers in Europe, comes from De Historia Stirpium Commentarii Insignes written by Leonhart Fuchs in 1542 (a mere 50 years after Columbus sailed the ocean blue), but he mistakenly thought the pepper originated from India. While his peppers may have come from India, India's peppers were most likely introduced by the Portuguese. Most of this paragraph is covered by Food in Early Modern Europe by Ken Albala.
So, food wise, not much of pepper is seen in the cookbooks of the seventeenth and eighteenth centuries. When this occurs, it's easy to proclaim "The poor popularized it", which is often the case in other foods (see tomatoes and potatoes as good examples). But in pepper's case, we can turn to medicine, specifically as documented in various pharmacopoeias. The druggists of those centuries popularized many "exotic" plants, and this included peppers themselves, with the capsaicin found within, and was used to settle the stomach.
What this means in regard to your question (as it relates to Italian cuisine) is that pepper's past is complicated. Pepper's past was propagated by the Portuguese, which leads me to believe that ports on the Piedmont Peninsula perceived peppers potential properly (sorry, I couldn't help myself). In other words, it arrived in the Italian region via the traditional routes/ports - Genoa, Venice, Palermo, etc. roughly around 1535 if my reading is correct. It's popularity was possibly helped by peasants, but one shouldn't discount the needs and influences of the pharmacists and doctors of the era either, nor their ability to increase demand. As for it's influence on cuisine, I'll go digging elsewhere
Most of this information was culled from Food in Early Modern Europe by Ken Albala, the Cambridge World History of Food, and various pharmacopoeias found on Google Books.