There are two ways that Divine Providence can affect, or interact with, the world. One way is for Divine Providence to emanate from the heavens without any prodding from the world “below”. An example of this is the miracle of the splitting of the sea. According to the Talmud the Jewish People didn’t merit a miracle. In fact they were hardly any holier that their Egyptian oppressors. God performed the miracle without any compelling reason from “below”.
The other way is for Divine Providence to be, so to speak, pulled down from below. An example of this is the miracle of Hanukah. The Jews formed an army and fought the Greeks and only then did God miraculously give them victory and freedom. The Jews “pulled down” the Divine Providence by starting the process of salvation themselves. The modern state of Israel might very well be another example of this idea of the Jews starting the process of salvation and, by doing so, pulling down the Divine salvation.
The revelation at Mt. Sinai was clearly an example of the first type of Divine Providence. God revealed Himself and the Torah to the Jewish People without any prodding from them. All they had to do was to be there and accept the revelation.
The Mishkan is an example of the second type of Divine Providence. By building the Mishkan and worshipping in it the Jews actively attempt to connect to the Divine and to bring the Divine Presence to dwell among the nation. The Mishkan then is certainly not a place for God to dwell in. Rather it is a vehicle for the Jewish People to “pull down” the Divine to dwell within each and every Jewish soul.
The Talmud calls the synagogue a “mikdash mi-at” or a small sanctuary. Just as the Mishkan, and later the Holy Temple in Jerusalem, was a vehicle for the Jews to “pull down” the Divine Presence so to is every synagogue a vehicle for the same. The synagogue is not “the” place where God dwells. God dwells everywhere or, as the Rebbe of Kotzk said, “God dwells wherever man lets him in”. Rather, the synagogue is the most opportune place for Jews to connect to God through prayer. Through our prayers we can draw the Divine Presence to dwell among us.