Politicians are elected to serve a specific number of years in office, which is called their term.
The process of campaigning/electioneering prior to the vote, and the vote itself, is called the election cycle. Like other cycles, it is a loop: campaign, election, swearing-in, legislating, campaigning again. In some legislative bodies, all of the politicians have their terms expire at the same time. In others, such as the United States Senate, the term lengths are all the same (six years) but they are offset so only one-third are up for election each cycle (every two years).
With all this in mind, "the 24 others who won their races in this cycle" means "the other politicians who were voted in to office this election year."