A "visa" is a document which permits you to visit or live in a foreign country.
Many countries have restrictions on who can live and work there. You might not be allowed to move there unless you meet some qualifications like being married to a citizen of that country, or having a job offer from a company which wants to hire you.
If a company wants to hire a foreigner, it must "sponsor" that person's visa application. This means that the company sends certain paperwork to the government to prove that it needs to hire this person. The company may have to spend some money on it. So a lot of companies do not sponsor foreigners' visas.
Some other things that companies "sponsor" include:
sponsor a TV show
sponsor a conference
In these cases, "sponsoring" something means giving money to support it.