When Jesus said this, he meant it literally. He did not mean that the meek shall find spiritual salvation or inherit heaven. These were addressed in the other beatitudes.
Why would the actual earth on which we live be eventually populated by the meek? The argument consists in two points. First, there is in general such a thing as progress. There are indeed numerous dead ends into which individuals and nations joyously directed themselves during the course of human history. But they lead nowhere. The sinners have to, on their own will and intellect and power, renounce sin, lest they never find happiness.
Here's a latest headline: "Islamic State Has Full Control of Syria's Palmyra." Here's the thing, though: there is nothing there but ancient ruins. ISIS is the undisputed ruler of a desert wasteland. And that is all this place will remain, until ISIS meeks up.
Second, the direction of social progress and justice points toward laissez-faire capitalism. But there is a eugenic tendency within the free market, as I have already pointed out, that causes those people who are more productive and better at satisfying consumer desires, i.e., at making other people happier, to "afford" more children than their less capable fellows. As a result, superior workers and entrepreneurs should be expected to leave more children who will, on average, inherit their benign powers.
Such productive men and women will be especially peaceful and more intelligent, including emotionally, in their inborn traits, (as being a successful entrepreneur entails reading one's customers) and will leave more children than the more aggressive and to that extent less useful to society persons. This, however, only given the laissez-faire ideology.
Therefore, if we want to fulfill Jesus' prophesy, then we'd better adopt unfettered capitalism ASAP.