H-1B Visa Sponsorship by Universities The statements of

&quot;Applicants who are invited to campus for interviews must be able to show proof that they will be eligible and qualified to work in the United States by the time of hire.&quot;

is almost associated with every open academic positions in US and I am wondering what does this statement clearly mean?
As far as I know, academic positions in US are opened to the scientifically eligible candidates worldwide, but receiving US work permit requires a sponsor (employer) inside US who should initiate visa application.
My 1st question: Does this sentence show unwillingness of the employer (university in this case) to support applicants' visa application? Does this position only open to those residing in US? or work permit holders?
From the immigration website, I learnt that there are some costs for the employer who is offering job to an outsider. But, universities are exempted from paying the biggest chunk ($1500).
My 2nd question: Considering low visa petition cost, what else is the problem that US-based universities are hesitant to hire such candidates.
My 3rd question: Is faculty selection in US-based universities based on candidate's merit or there are other parameters when selecting them.
Edit: I think the email received from UW can be of help to get better idea of the situation.

Dear XYZ,
When completing your application for employment at the University of Washington you were asked to complete the US Work Authorization Assessment. The second question asked was:
To legally work in the United States, will you require University of Washington sponsorship for an H1-B, other UW sponsored visa, or UW sponsored green card now or in the future?
You responded yes, that you would need sponsorship, either now or in the future. As a result you are not eligible for employment in staff positions at the University.

I emailed to HR of UW for clarification with this note:

Does UW consider only those foreigners for staff position who are currently having work permit in US? Those who are willing to obtain work permit based on the UW job offer are ineligible to work in UW?

UW Reply (Though I think the reply is by a robot):

Unfortunately, the UW does not provide sponsorship for Green Cards( or HB-1 visas, or any other type of work authorization requiring sponsorship.) for staff positions.


EDIT 2: Though my question is about academic staff position, about university of washington, I applied for research scientist. Here is the link to the job opening.
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.