Julius and Ethel Rosenberg were American citizens who spied on the United States on behalf of the USSR, giving confidential information about jets, radar, and more.
This attracted much attention as the United States was in the midst of a "red scare" where everyone was afraid of communist infiltration. The Rosenbergs gave tangible proof that the USSR was in fact infiltrating the USA.
In this speech, Lyndon Johnson also discussed racial inequality, which was rampant in the 1960s and causing outcries from people who wanted drastic change.