Where do I find at what time an airport was closed, due to inclement weather, on a specific date? I completed the first leg of my flight, which was one hour from take-off to landing. When I arrived in my connecting city, my final destination airport was closed due to weather conditions. Why did they not cancel my entire flight before I departed on the initial leg of the flight?
The best way to get that information would be to call Airfield Operations at the airport in question. They can tell you what time the airfield was closed, IF it was the airport itself that was "closed" in that situation. Oftentimes, however, the field itself isn't technically closed, but the weather was simply so bad that nobody was getting in, and that gets discussed as "the airport is closed" due to weather. Not technically accurate, but close enough for discussions of "can flights get in there or not."
As for what happened before you left for your connecting city, all sorts of answers are possible. Perhaps the weather wasn't that bad initially, and it looked like your connecting flight had a good chance of going. Or perhaps "the system" slipped up, and they should have notified you that the connecting city was as far as you'd be likely to get, and they didn't. Or perhaps there were PA messages made to that effect that you didn't hear. (It happens.)
Also, an airline wouldn't cancel the flight to the connecting city, since some passengers presumably were trying to get there as their destination, and others had other connections to make to cities that weren't affected. What could happen if they knew early enough that your second flight wasn't going to go, and recognized soon enough that "this passenger won't get where they're going today", is they'd advise you of the situation, and offer you the option to cancel your trip, rebook the next day, or get to the connecting city so you're that much closer when the weather has passed & flights resume. Depending on multiple factors, any of those might be a reasonable choice for a passenger to take, so they offer the options, rather than unilaterally cancelling your ticket.
It's also possible that the airport itself made some sort of call that "we're closed" at some point after your first flight had taken off, surprising the dispatchers at the airline. In that case, you are, plain & simply, out of luck. Not desirable, but in the millions & millions of airline flights and hundreds of millions of passengers flying every year, it sometimes happens.
Reviewing the weather reports for that timeframe from someplace like Weather Underground might shed light on just how bad the weather was, and when the forecasts were amended. Making sense of that would require some familiarity with aviation weather reporting formats, though.