Why are some countries more popular destinations for higher education than the other?
Why do some countries get more international students than others? Wikipedia gives a few reasons:

Traditionally the U.S and U.K have been the most prestigious choices, because of the presence of top 10 rankings Universities such as Harvard, Oxford, MIT and Cambridge. More recently however they have had to compete with the rapidly growing Asian higher education market. While US is the leading destination for foreign students, there is increasing competition from several destinations in East Asia such as China, Korea, Japan and Taiwan which are keen to attract foreign students for reputation and demographic reasons.

So university ranking is a major reason. Other reasons implied by this paragraph include competition: because international students pay full tuition fees, they are very lucrative, and universities want them. Therefore they send representatives worldwide to recruit students. Government incentives, e.g. less restrictive visa requirements, certainly help too.
Otherwise there is:

Due to the research base and funding opportunities, the USA was one of the key destinations of choice, especially in science and technology at master’s and doctoral education levels (National Research Council 2005).

Given that &quot;Students were motivated to study abroad as they were also getting economic rewards from the skills gap they could fill in high-demand, technology-related jobs&quot; (quote taken from the same source) it's natural that the countries with the most research funding gets the students. Historically, that is the US.
That explains much of your countries:

Ireland, New Zealand &amp; South Africa don't have highly-ranked universities.
Switzerland actually has a high number of international students at tertiary level - 16.9% of tertiary students are from outside the country. It's not in that list because there aren't many tertiary students in the country in the first place.
China's research funding has been rising tremendously, plus there are top universities in the country.
Japan's R&amp;D funding is also one of the world's highest, plus there are top universities in the country.