Refers to a person who agrees with or supports a particular political or ideological group, often without being an official member. The term can carry a slightly negative nuance of passive agreement.
He is not a party member, but he is active as a sympathizer.
That organization has many sympathizers.
Compound of 同調 (dōchō, 'agreement; conformity') + 者 (sha, 'person'). The derivation is straightforward from the component kanji meanings.