OK, its a Kwikset..or KW TYPE. the latch part that is IN THE DOOR, not visible here, but between this hole and the edge of the door has broken internally the lock part you are holding should with a slight turn come out. if it does not, you will end up taking the door off its hinges, and the 2 round parts that you are holding, will need to be broken off so that the bolt can be removed from the edge of the door., then replace..
BTW, this HAPPENS, had to do a few myself like it. in ANY CASE, you will have to have the door OFF the hinges for a bit to get this piece out --Shiva--