It was more complicated than just beans, as Umberto Eco himself says in your source. He mentions political reforms, the growth of the German Empire, the invention of a better horse collar and metal horseshoes, the spread of the plowshare. All of these things had an impact on agricultural productivity, and therefore the land's capacity to support people. And in addition to all that, yes, there were beans.

In general, we have good evidence that Europe's population increased quite dramatically during the High Middle Ages. Between the time of Charlemagne to around 1250 AD, the population of Europe doubled in most places, and tripled in some. Obviously, more people eat more food, so there must have been a corresponding growth in food production. Where did it come from?

Political stability played a big role. The end of the Viking raids and the growth of large kingdoms and empires encouraged people to spread further, and huge tracts of untouched land were brought under the plow. This period is sometimes called "the Great Clearing", as much of Europe's forests were chopped down to make room for farmland.

The technological improvements Eco listed also played a large role in increasing productivity per acre. One thing he fails to mention is the Medieval Warm Period, a 300 year phase of warm and stable climate across most of Europe between 950 AD and 1250 AD, which coincides with the period of high population growth. During this time, the growing season was extended, crop harvests increased, and a significant amount of marginal land became suitable for agriculture.

The end of this warm period coincides with a number of significant famines during the late Medieval period, probably because the population had grown to the limit allowed by food availability, and any dip in the food supply meant that people starved. Then came the great plague, and Europe's population dipped down quite dramatically for a short while.

So where are beans in all this? One thing to remember is that veggies were not quite the same as we know them today. Cabbages grew in small loose bundles of leaves, there were no "heads of cabbage". Lettuce was about the third of today's size, small and bitter. Carrots looked more like parsnips, more woody, less sweet. Most veggies were much closer to the wild form than they are today. But in this vegetable-poor landscape, cereals and legumes were still highly nutritious. Of course productivity per acre was much lower than today, but the wheat, barley, oats they grew were still high in nutrients, especially in comparison to other stuff they were growing.

Charlemagne's reforms included prescriptions for what to grow, and the long list does include peas, chickpeas and kidney beans. Although the type of legumes grown varied regionally across Europe, we know that in general, the overall quantity of legumes grew quite significantly during the High Middle Ages. For example, here's the situation in England:

• The contribution of legumes to the diet of English peasants and farm servants, c.1300. La Poutre Hugo JP (June 2015) The Agricultural History Review 63(1). Full PDF here.

There are lots of lines of evidence, such as the total acreage planted with legumes, or the fraction of legumes distributed to agricultural workers by the landowner as part of their remuneration. It tends to show that while the diet of the nobles didn't change all that much (they got a much larger fraction of their protein from animal sources), the diet of the peasants changed to include more legumes. This was true over much of western and southern Europe, and probably to some degree in eastern Europe as well.

Back in those days "peas" didn't just refer to the peas we buy in grocery stores today, it also included a number of pea-related plants that produce similar seeds. Typically, they were dried, and in the dry form could easily be stored for the winter. The usual way to eat them would have been as "pottage", which would be a mix of oats (the cheapest cereal at the time) and dried peas, boiled together in water with seasoning and any other available veggie added.

Peas were also ground into flour, and this was mixed with barley or wheat flour to bake bread. The cheaper (and coarser) kinds of bread included some significant amount of "pea flour", which made it quite nutritious.

Farmers quickly learned that legumes were a great way to revitalize the soil, since they harbor bacteria in their root systems that can fix nitrogen. This pretty much removes the need for any additional nitrogenous fertilizer, if you include peas in your crop rotation.

So, in short, yes, the population of Europe did increase during this period, and yes, there was a concomitant increase in the consumption of beans, especially by the poor. It seems likely that there is some association between the two, but giving beans all the credit would be simplistic. In particular, the effect of the Medieval Warm Period can't be underestimated, along with societal factors (political stability allowing the clearing of more land for farming) and technology (better horse collar, iron horseshoes, the spread of plowshares).