Backset is the acidic leftover liquid in the still after you have run it. We use it for PH and some flavour, if you pull it straight from the still while it is hot, it is also good for melting sugar and softening ingredients / grain etc.
If you are doing a 20L wash it is calling for 4L of the backset plus the ingredients and fill the remainder with water.
The slurry in the bottom is trub and is made up of mostly yeast. If you are wanting to reuse this you will need to use a 2nd bucket to keep it separate until the wash is cool enough not to kill it. After 5 or so generations you will likely need to repitch new stuff anyway.
The other option (which i do) is to add the hot backset to the fermenter and add new yeast once it has cooled. The dead yeast will become nutrients for the new stuff. Yeast is cheap.