Here's my understanding of the strategy:
-putting in a DACA fix has broad bipartisan support. Nearly everyone who matters in both parties has, at one point or another, come out in favor of a DACA fix at least in principle (Republicans generally want it tied to border security funding and/or the wall; Democrats want a "clean" bill). Even Paul Ryan urged Trump not to cancel DACA in the first place. But he did, and here we are.
-DACA specifically, and immigration more generally, are white hot issues for the Democratic base right now. As the party out of power, Democrats do not have a lot of leverage. This is pretty much it for them for the foreseeable future. They have decided that they want to use that leverage on DACA because maintaining enthusiasm among their constituency is critical to the upcoming midterms.
-The Democrats are willing to bet that because Republicans hold the White House and both houses of Congress, everyone but the die hard Republican base will lay the political blame on the Republicans. In general, the party of the President gets blamed when this stuff happens. Apparently Trump himself said during the 2013 shutdown that "The president, in all fairness, he's the leader, he's the one that has to get everybody in the room and get it done. So he does have a lot of pressure to get this problem solved. He's got a big problem." The last time there were shutdowns and unified control of WH and Congress was during the Carter administration.
It's a gamble--and the Republicans are out in force messaging that Democrats are to blame--but that's the gamble the Democrats are making.