The team had been called the Reds since their admission to the National League. The “Red Scare” of the 1950s, however, made the owners nervous about having a team called the Reds. So they added “legs” to the moniker, because apparently Redlegs were clearly NOT communists, whereas as Reds were suspect (their thinking, not mine).
The name never particularly caught on and they reverted to the Reds by the end of the decade. You can read more here. Their mascot, however, is called Mr. Redlegs.
So both these teams have had name changes in response to various pressures.