Why was Gerald Ford picked to be Vice President when Spiro Agnew resigned?

In the early years of the American Republic, the job of the Vice President of the United States was considered so insignificant that several potential candidates turned the job down when nominated: eighteen American presidents served without a vice president. Over time, the job took on new meaning as the government became more complex and some presidents allowed the vice president to assume more critical duties. Added to the improved status was the idea that the job of vice president was a natural path to the becoming President of the United States.
Fourteen presidents were former vice presidents, with five of those being elected and eight serving when the president died in office. Only one assumed the office after the resignation of the president: Gerald Ford became President of the United States when Richard Nixon resigned. Ford's rise to the highest office in the land was a combination of unique events, some of which took place many years before Nixon became President of the United States.
The Twenty-Fifth Amendment, Section 2 states, "Whenever there is a vacancy in the office of the Vice President, the President shall nominate a Vice President who shall take office upon confirmation by a majority vote of both Houses of Congress." Gerald Ford was nominated and confirmed by both Houses of Congress to replace Spiro Agnew as Nixon's Vice President and eventually replaced Nixon as President.
Here is the historical irony of the Twenty-Fifth Amendment concerning Nixon. The amendment passed after the assassination of President Kennedy. Kennedy defeated Nixon in the 1960 presidential election. Appearing before a Senate Subcommittee on Constitutional Amendments, Nixon argued that the Electoral College results should be used to select a Vice President in the event that a vacancy occurred.
Up until 1962, the order of succession was the Speaker of the House and President Pro Tempore of the Senate. In Nixon's view, this was a problem, since either person could potentially be a member of the opposite party and would therefore place a partisan in an office which, increasingly, was evolving into a position with significant agency to carry out presidential policy. Nixon therefore proposed that the Electoral College elect the next Vice President, thereby assuring a member of the same party would assume the role.
The Senate, as is often the case in political matters, rejected the advice of Nixon and instead adopted the language found in the Twenty-Fifth Amendment. In addition to clarifying what happens when the Vice President is unable to serve, the Senate tweaked the amendment to include the language of what happens when the President is temporarily unable to serve (illness, surgery, etc.). The Twenty-Fifth Amendment was ratified in February of 1967. The irony, of course, is that Gerald Ford rose to become President of the United States under the additions to the amendment which Nixon had argued against.
The answer to your question is that Gerald Ford was well liked by his colleagues in the House. Ford was a Republican and the House Minority Leader. He was considered to be a man of principle, but willing to make a compromise. Ford was calm, experienced, and non-controversial, which made him the perfect pick to counter the volatile Nixon in the throes of a scandal. Former Vice President Spiro Agnew was being investigated by the United States Attorney on suspicion of criminal conspiracy, bribery, extortion, and tax fraud. He was accused of soliciting and accepting kickbacks from contractors during his time as Baltimore County Executive and Governor of Maryland.
Gerald Ford became Vice President indirectly as the result of the assassination of John Kennedy and the United States Senate amending the Constitution to formalize a process for choosing a Vice President in the event one was needed. The Senate rejected Nixon's proposal in favor of the Amendment, and it was Nixon that Ford eventually replaced. The irony is that distrust of Nixon played a role in Ford's rise to both offices.
https://www.law.cornell.edu/constitution/amendmentxxv

https://www.whitehouse.gov/about-the-white-house/presidents/gerald-r-ford/