I'd pronounce it
"Weeknights at 11. 10 central."
What it means -
The US (and Canada) are spread across several time zones. When TV first started out (in the 1940s and 1950s), everything was done on New York time. So a show would be at 8pm in New York, but 7pm in Chicago, and then 5pm in Los Angeles. (or LA would get the same show a day later.) not that many people had TVs, the local stations had a lot of independence.
As more people had tvs and the schedules became more standardized (and technology got better), things changed. And instead, they showed the programs at 8pm everywhere - except for the central time zone. In that time zone, everything still shows at the same actual time as the east coast time zone (which is an hour earlier on the clock).