Okay, so the thing is...that this isn't really true. Kind of. But it kind of is. Kind of. The problem was not so much an imminent overpopulation crisis, it was 1) autarky, 2) plans for sizable population expansion in the future, and 3) a need for a casus belli that will motivate the populace. I will primarily speak for Germany, because I am more familiar with Germany than other sources and, as such, I can talk about them off the top of my head far more accurately than the others.

So, autarky. Autarky is the idea that a nation should be completely self-sufficient, that a nation should be able to sustain itself entirely through it's own means without any outside help. As a rule, autarky has pretty much never been a great idea, more so in the modern era than the past, but still in the past as well (primarily because it's just really, really economically inefficient). Now, to be fair, Germany, at the very least, although I cannot speak for the others, had a rather good reason for embracing some autarkic ideas after WWI, and that was because the British blockade had absolutely devastated Germany and caused a lot of starvation throughout the war, especially during the middle and late periods. The inability to import sufficient food during the war killed a lot of Germans and also wore down morale and political will to continue the fight. The global economic downturn caused by the Great Depression, also likely contributed to this rather sizably (especially for Germany), although I cannot say how much with any fair degree of certainty so I won't.

Secondly, the desire to grow the population. I am going to talk specifically about Germany here because I cannot really speak for the others. That being said, I would also assume, keyword here being assume and not know that all countries wanted to grow their population sizably because, in large part, a larger population is always good for your economy. The decline of population growth rates in France while the opposite happened in Germany was a big part of the reason why France waned as Germany rose - and it was also an issue that France tried to combat throughout the 19th and 20th century. Since we are discussing Germany, I am also going to include ideology and policy in this in since, when discussing the Nazis, as you'll soon see, land is going to be really, really important.

So, one thing to note about the Nazis is that although the popular imagination always portrays Nazi Germany as this great industrial power house full of cutting-edge technology, which to be fair to them, they were a very advanced nation with some of the most productive industry in Europe, the Nazis themselves were, from an ideological standpoint, anything but. They were a very conservative, pro-agrarian, pro-rural society - the irony of us, today, remembering Nazi Germany as an urbanized industrial powerhouse instead of a rural breadbasket speaks wonders of how ineffective the Nazis were at the whole agricultural policy thing (in fact, under them agricultural yields dropped rather consistently, farmers frequently got shafted, and Nazi Germany not only had frequent problems with agricultural production, but also had constant complaints from farmers about how they, the farmers, were just continually getting the short end of the stick) - but I'm getting sidetracked. So, for the Nazis, the traditional life style of large, rural families with their own family farm was the ideal. . . Buttt, the problem was that if you insisted on having large families and everyone ending up with a farm, then you were going to need lots and lots and lots of land, at least eventually. So for Germany a large issue was literally just having enough land to make sure that you could have huge families continuing to split off and form their own family farms.

So, yes. One factor was autarky and being able to be completely self sufficient and, as such, having all the natural resources and food you would ever need. Another factor was having the land to do so in, the land to expand your population in, and also the land to put your rather ridiculous and inefficient ideologically-based social and economic policy into place. And the last reason that I'll touch on is as a casus belli. As such they were very flexible

These terms and ideologies also serves as a blanket casus belli for these nations. This is little different than Hitler's use of comparatively small chunks of German populations existing in foreign countries as a pretext to "protect" those people and thus demanding land, under the threat of invasion, to do so. It served as a blanket excuse to justify war for expansion under a reason that some people could get behind, especially those who had suffered starvation due to the British blockade or the Great Depression. And these nations had ideas for what they wanted their nations to look like. In the case of Germany, for example, Hitler wanted to reclaim pretty much all "German" lands, which he seemed to define, rather arbitrarily as pretty much any land that had been part of the HRE (with exception to the Italian lands, so places like the Lowlands and the Bohemian lands of Austria/Austria-Hungary) or had even relatively loose connections to German-ness (like the Nordic countries). In other words, as with most other things, the Nazis exercised a great deal of arbitrariness and running fast and loose with history and reality where it suited them. In the case of Italy, of course, we often hear of them wanting to recreate the Roman Empire. In the case of Japan, we had the Greater East Asia "Cooperation" Sphere (ironic quotes added by moi, because it was about as "cooperative" as the German occupation of Poland). And while the borders imagined by these nations certainly grew over time as they saw great successes during the early stages of the war, they were simply expanding on expansionist ideas that had existed pretty much from the get-go.

Sources (primarily on the points about Nazi ideology and agriculture):

Neil Gregor's, Nazism

Benjamin Sax & Dieter Kuntz, Inside Hitler's Germany: A Documentary History of Life Inside the Third Reich