The word synagogue comes from a Greek word meaning "assembly of men" or "congregation," and it was used much like the English word "church." The "synagogue of Satan," then, is an assembly or congregation—a church—made up of the individuals who "say they are Jews, and are not." The term "Jew" is used here in a spiritual sense.
Notice the apostle Paul's definition of a spiritual Jew in Romans 2:29: "He is a Jew who is one inwardly, and circumcision is that of the heart, in the Spirit, and not in the letter; whose praise is not from men but from God." In this case, it is not one's physical race that counts, but his spiritual condition (Galatians 3:27-29; Romans 4:16). True Christians are spiritual Jews because Jesus says in John 4:22, "Salvation is of the Jews."
Those in the "synagogue of Satan" say they are spiritual Jews—pretend to be real Christians—but are not. This false church was already developing in the days of the apostle John, masquerading as God's true church. It had congregations in the cities of Smyrna and Philadelphia in Asia Minor even at that early time. It and its daughter churches are further described in Revelation 17.
One of the hallmarks of Satan's activities is deception, particularly counterfeiting the things of God. "For such are false apostles, deceitful workers, transforming themselves into apostles of Christ. And no wonder! For Satan himself transforms himself into an angel of light. Therefore it is no great thing if his ministers [servants] transform themselves into ministers of righteousness, whose end will be according to their works" (II Corinthians 11:13-15). Jesus says we will know such false ministers by their fruits (Matthew 7:15-20). Thus, we should make sure that the church we attend and its ministers are following biblical doctrines and producing the fruit of righteousness. Otherwise, it may well be a "synagogue of Satan."