What does mean work eligibility?

Any employer is going to want to see that you are eligible to work in the US or the UK. In the US at least sponsoring a foreign employee to ensure their eligibility is expensive and not easy to do.

Essentially (again, for the US) you'll need to show a work-eligible visa or proof of citizenship. Any document you have that proves you are currently legally allowed to work in the US. It doesn't necessarily have to be a visa, but essentially that's what they're asking for in your situation.

Eligibility Meaning

It mean are you legally able to work in united states

If you have a student visa, you are not allowed to work in the US. You must have what they call a green card. It is a permit to allow someone who is not a citizen to work. Employers are starting to ask everyone for proof. Also you can show a Social Security card and a Drivers License. You need two forms of US government identification.