H-1B Visa Sponsorship by Universities
My suspicion is that this phrase is primarily to cover the university legally in case the person hired in fact does not have the eligibility to work in the US. In such a case, without this clause, the person could make the argument that the university is in breach of contract for not hiring them and doing whatever is necessary in order to make them eligible. But IANAL !

In general I'd imagine only very small universities or places with limited resources would have trouble sponsoring a candidate for an H1-B. Certainly not a university like UW. Most universities are quite willing and able to sponsor new applicants for H1Bs and green cards. 

The correct interpretation here is I think as is mentioned in the comments: the candidate should be eligible to work in that they can get an H1B visa.