Well, we get this expression from "watertight" or "airtight".
A watertight container doesn't let any liquid in (or out).
An airtight container doesn't let any air in (or out).
If you're wondering, these words are now familiar enough that we don't need a hyphen (originally, we wrote "air-tight").
So, a day-tight compartment doesn't let any influence from yesterday or tomorrow in. Also, whatever happens on one day isn't allowed to affect tomorrow.