What to do about a cracked stair when I can't find how the stair is held in place?
If we begin with the assumption that the broken piece is to be replaced, then anything that involves destroying only that one piece is fair game -- unless you'll be wanting to create a careful replica of the broken piece there's little reason to restrict yourself to removing it intact. Open up a hole in that plank so that you can feel and then see into the space and you'll gain understanding of how the builder assembled the stairs.
To open such a hole consider a saw that can do a &quot;plunge cut.&quot; This means it can cut into the surface of a board as opposed to beginning at an edge. A reciprocating saw with a hook-tip blade can do this, as can an oscillating multi-tool. Another option is to drill a hole through the board and then use a jigsaw to enlarge the hole.
Things to watch out for:

The tread boards could be tongue-and-groove. If so it'll require some care to remove the rear board without damaging the front board.
The riser board is likely to overlap the broken rear tread board. You might have to cut the tread board into pieces that can be slid out from beneath the riser.

As you remove more pieces of the tread board it'll become more clear how the board was secured and which other pieces, if any, will have to be removed to make a good repair possible.