In the dictionary of etymology it says that this usage of the verb dates back to 1943. What happened in 1943? Was there a new dictionary published? Or how else was "to screen" officially repurposed?
In 1943, the Baltimore Sun published the following in an article:
These offices ‘screen’ a list of prospects for the employers.
At some point in time, this quote was found by the Oxford English Dictionary (or someone sent it to them). This quote can be found in the second edition of the Oxford English Dictionary (OED2), published in 1989. The OED2 is one of the main sources for Etymonline, which is what I assume you mean by "dictionary of etymology".
But the date's not very significant at all. Some words have a definite date of origin (e.g. "cromulent"), but many—the vast, vast majority—only have a first known use. For "screen" there are only first known uses of different senses. Since 1989, numerous different websites allowing you to search books, newspapers, etc. have popped up, making it orders of magnitude easier to find earlier examples vs. what is found in the OED2.
The story doesn't stop there for this word. In March 2017 the OED published an updated page for "screen" which features new, earlier examples. Here's one from the 1942 Cedar Rapids (Iowa) Gazette:
USES work applicants are being carefully ‘screened’ by test to determine their abilities.
And another source also from 1942, from the Public Admin. Review referring to a thing instead of a person:
By means of this central index it is possible to ‘screen’ applications against available information and to review data for violations much faster than would otherwise be feasible.
If we're looking at screening for diseases, the earliest source given by the OED is 1938 from the Journal of the Royal Statistical Society:
A coefficient of correlation..provides a measure of the general association of an index with clinical assessment, whereas practical interest is directed on the particular boys whom it is desired to screen.
(In the medical sense, the noun "screening" dates back even further, to 1920.)
In the future? It may be possible to push the date even earlier. It may be possible to do that now even, considering that it's easy to miss things when a word has many different meanings.
All in all, it's a pretty natural extension of meanings that already existed for "screen", such as sieving.
The meaning of the verb screen, to select, reject, consider, or group by examining systematically’, begins with a quite literal screen used to sort or size any number of products:
He declared that, if the Pittsburg operators can not sell run-of-mine coal on the lakes, in competition with other fields, they will have to screen the output and ship only the nut or lump fuel to the competitive markets. — Indianapolis News, 30 Jan. 1900.
The next step is to extend the process metaphorically, in this case, by a simile:
The Governor charged that “these committees apparently operate as a screen to sift applicants for commissions,” although “no publication of their functions and methods has been made, and the public in general is unaware of their activities.” — The Evening Star (Washington, DC), 21 Mar. 1942.
The screen here is still a literal one: the committee “operates as a screen;” they are not yet screening applicants. This step, however, was not long in coming:
Continuing their efforts to cut gasoline consumption here by 267,000 gallons a month, District OPA officials have sought the assistance of the Army and Navy in screening out certain types of applicants for preferred mileage, it was learned yesterday. — The Evening Star (Washington, DC), 31 Oct. 1943.
In other words, we will have to screen applications very carefully and continue to depend upon recaps and Grade III in many instances. — Steamboat Pilot (CO), 30 Sept. 1943.
What I did point out was, that in view of the fact that at present there are many more applicants than there are blocks available, it would have been advisable to screen applicants a little more thoroughly, and to confine them, mainly, to those who had previous agricultural experience. — W. A. Bethune, Letter to the Editor, The Mercury (Hobart TAS), 21 Oct. 1947.
Once a new figurative use becomes established, the metaphorical force that created it completely recedes. No employer or university admissions officer who screens applicants/applications today is going to give a single thought to sorting coal.
The 1943 date is the first appearance in print of the new usage, which many speakers during the Second World War apparently considered an apt way of describing various selection processes during a time of civilian rationing and military recruitment. Dictionaries will eventually acknowledge a new meaning of a word once enough elite speakers use it. Dictionaries record — but do not create — language. People do.