The word "causing" is doing an awful lot of work in that sentence. Dems didn't "cause" any of this.

They would have "caused" a shutdown if they voted against the CR.

Democrats have no control in the House. They are the minority party. The majority party can pass any bill it wants, and the minority cannot stop them.

The Democrats do have some measure of control in the Senate. The Senate rules allow the minority party to stop virtually all legislation, other than reconciliation bills (or if the majority has a supermajority). The Democrats do have some power in that chamber.

With power, comes responsibility for how you exercise that power. The Democrats in the House could freely vote against the CR without much responsibility for what happens if it fails, because they have very little power - and therefore virtually no responsibility - in that chamber. But that's not true of the Senate. There, the Democrats would have had primary responsibility for a shutdown - because they have the power to force that outcome, and it would happen because of their choices.

There was a play to stop the CR - hold the line in the House and hope that the internal tensions in the GOP prevented them from getting to 216. The GOP would have borne the primary responsibility for the bill failing in the first place. But once that failed, the Democrats no longer had an option where the CR could falter and the resulting shutdown would not be due to their choice.