Why are there so many migrants in the modern world?
Are current flows of migration bigger that they were in the past?

We tend to forget history because the numbers involved currently are smaller 
relative to world population. For instance, the 8.8 million immigrants who came to the United States in 1901-1910 period is of similar magnitude to the 9.1 million immigrants who came in the 1991-2000 period, but both the U.S. and world populations were much larger than in 1901-1910.

Richard B. Freeman in a nice paper, entitled "People Flows in Globalization," reminds us about this fact. 

Some countries indeed receive large inflows of migrants

The US, Russia and Germany are the biggest recipients of immigration (see Table 1 in the above paper). These flows have consequences for factor endowment and there is considerable and often heated controversy over their impacts on destination countries.

How Do Immigrants Affect Their Countries of Arrival?

Freeman examines this question (p.18 to 23) and reviews the empirics. In the basic model of immigration, immigrants reduce earnings of substitute factors and raise the earnings of complementary factors, where complements might include capital and some types of native-born labor. In the basic model, there is full employment but you may intuitively consider quantitative adjustments instead of price/wage adjustements by introducing frictions in the labor market. Note than in most advanced countries, 
immigrants have higher unemployment rates and lower employment rates than native workers.

Related to this question a classical paper is "The Economics of Immigration," with theory and evidence by Georges Borjas in Journal of Economic Literature, December 1994. More recently, in 2015, Borjas published an interesting review on Immigration and Globalization. You mal also be intereted in reading his paper on how the collapse of the Soviet Union affects the productivity of American mathematicians.

Regarding policies Freeman  considers "radically economic policies" such
as auctioning immigration visas or charging sizeable fees and spending the funds on current residents to increase the economic incentive for advanced countries to accept greater immigration.