Firstly, the answer will tackle the question's false assumption that the US is the most attractive destination for international students. Secondly, I will cite some of the factors making a country/region's education system attractive to international students. Finally, to tackle some of the comments, I will present a chart showing number of international students per capita in selected countries.
USA is not the most attractive destination for international students
According to the OECD Factbook 2011-2012: Economic, Environmental and Social Statistics, the number one destination of foreign students among OECD countries is Europe followed by Northern American region:
European countries in the OECD were the destination for 38% of foreign students in 2009 followed by North American countries (23%). Despite the strong increase in absolute numbers, these proportions have remained stable during the last decade.
To put the numbers above to global perspective, observe also that
Foreign students enrolled in G20 countries account for 83% of total foreign students, and students in the OECD area represent 77% of the total foreign students enrolled worldwide.
Factors driving attractiveness of higher education in OECD countries
Again, according to the same source (emphasis added):
Language as well as cultural considerations, quality of programmes, geographic proximity and similarity of education systems are determining factors driving student mobility. The destinations of international students highlight the attractiveness of specific education systems, whether because of their academic reputation or because of subsequent immigration opportunities.
Commenters to the question cite the ratio of international students per capita as an indicator of attractiveness of education system for foreign students. While I do not see any direct correlation between attractiveness of an educational system and the ratio of foreign students per capita (countries can be arbitrarily protective, or non-protective w.r.t. their own citizens), I prepared the following chart from the OECD data (relevant to year 2009):
The chart was constructed by merging data from the OECD.Stat with OECD countries population data from OECD population 2009 as published in the corresponding section of the OECD Factbook 2001-2012. The computation is done on non-citizen students column for the year 2009, except for United States it is the number of non-resident students (due to lack of a non-citizen students datapoint).