The mid-year convention (or mid-year discounting) treats cash flows as if they are all earned at the midpoint of the year as opposed to all at the end of the year (which would be known as period-end discounting).
We would use the mid-year convention in a DCF because when you treat cash flows as if they are all earned at the very end of the year, you technically over-discount them in your DCF.
To avoid this over-discounting phenomenon, we use the mid-year convention.
Note that you should really only use the mid-year convention if you assume that the cash flows are evenly distributed throughout the year. For highly seasonal businesses, it may not make sense to use the mid-year convention.