(This is a very US-centric response given the way OP phrased their question, but there are fascinating threads elsewhere in the world, such as the civil service exam system in China, that I hope others can address.)

What most people would consider modern selection practices for jobs originated in the late 19th century. We credit the early psychologist James McKeen Catell with originating the concept of a "mental test" in the 1890s and offering the conjecture that mental test scores could predict future performance in job and academic contexts. Hugo Münsterberg, who had vast interests in applied psychology of all kinds, advanced this line of thinking with his work Psychology and Industrial Efficiency (1913). This text offered the first complete thinking about selecting people for the work to which they are best suited, and likely to be successful, based on their abilities and attributes.

Katzell and Austin (1992) identified four major non-psychology forces that emerged between Catell's early work and the outbreak of WWI that fueled massive growth in personnel selection practices:

• Basic advances in statistical concepts that made forecasting performance via statistical inference possible.
• Increasing acceptance of Darwinian evolutionary concepts, including the bastardized application of natural selection to personnel selection.
• Growing American mythologies around meritocracy and work ethic that fueled a desire to identify "deserving" candidates for jobs.
• The emergence of management as an intellectual discipline, driven in large part by Frederick Winslow Taylor's work on "scientific management" that was highly publicized to the general public.

The outbreak of both World Wars was also massively important to advancing modern personnel selection. Both war efforts drew on the talents of applied psychologists to develop new assessments and solve practical problems of selecting, placing, and organizing huge numbers of people to create efficient and effective military and civilian forces. Critically, both wars also normalized selection by exposing recruits to these practices, which subsequently enabled those practices to easily spill out into the postwar public and private sectors with little resistance from applicants. For example, the first cognitive ability test used for widespread personnel selection was the Army Alpha (and Beta, the parallel version for illiterate recruits) developed by Robert Yerkes and the Committee on the Psychological Examination of Recruits. This is a terribly imperfect test in a great many respects, but it was the first systemic tool used to identify Army recruits who were suitable for officer training.

Selection has come a long way since the late 1940s, but immediate post-WWII selection practices would be recognizable to most people today. By that point, tools such as interviews, work samples, simulations, and tests/questionnaires were adopted by large businesses, such as AT&T, that eagerly hired many of the applied psychologists who worked on wartime selection problems, such as Donald MacKinnon (whose work on selecting spies for the nascent OSS/CIA is both fascinating and hilarious). These practices became benchmarks that other businesses rapidly adopted.

Katzell, R. A., & Austin, J. T. (1992). From then to now: The development of industrial-organizational psychology in the United States. Journal of Applied Psychology, 77(6), 803-835.

Koppes, L. L. (Ed.). (2014). Historical perspectives in industrial and organizational psychology. Psychology Press.