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The two letter codes were standardized at the same time ZIP was introduced. From USPS:

Until 1963 the Post Office Department preferred that state and territorial names be written out in full to avoid confusion, but accepted the popular public practice of abbreviation.

The Department published a list of preferred abbreviations in the 1831 Table of Post Offices in the United States. The United States Official Postal Guide, first published in 1874, also provided lists of preferred spellings. Most of the abbreviations in use in 1874 were still in use in 1943, although new states and territories had been added, and the abbreviation for California had been lengthened, from “Cal.” to “Calif.”

On July 1, 1963, the Post Office Department implemented the five-digit ZIP Code, which was placed after the state name in the last line of an address. To provide room for the ZIP Code, the Department issued two-letter abbreviations for all states and territories. Publication 59, Abbreviations for Use with ZIP Code, issued by the Department in October 1963, told why. Two-letter abbreviations were:

. . . provided by the Post Office Department as an aid to mailers in accommodating ZIP Codes within the usual City-State line of addresses. The abbreviations are based on a maximum 23-position line, because this has been found to be the most universally acceptable line capacity basis for major addressing systems. A breakdown of the City-State-ZIP Code line positions is as follows: 13 positions for city, 1 space between city and State designation, 2 positions for State designation, 2 spaces between State designation and ZIP Code, and 5 positions for ZIP Code.

To date, only one change has been made to the abbreviations issued in 1963. In November 1969, at the request of the Canadian postal administration, the two-letter abbreviation for Nebraska, originally NB, was changed to NE, to avoid confusion with New Brunswick in Canada.