How to play the level in the official community site archive?

The principia I played was downloaded from this site, but when I tried to play the level of the official community site, the principia did not respond.What should I do?

From the downloads page:

Hey! This download page is from before the source release and contains modded versions of the old proprietary 1.5.1 intended for use with principia-web. There are not publicly available binaries of the open source version yet, but you can currently compile from source for both Windows and Linux.

The official community site archive only works with the open source version, the current downloads are of the old proprietary 1.5.1 patched to connect to principia-web. I've been meaning to update them with 1.5.2 beta builds but there are currently some issues with the Android version I haven't yet gotten around to fixing.

But can the open source version of principia open the level of this website?

Yes it's going to be able to open levels from both, so there's a primary community site that it will connect to for things such as uploading, authentication and playing levels when a domain is not specified (by default principia-web.se). But in addition it can connect to any other site domain (e.g. archive.principia-web.se, any other site that would want to host and distribute Principia levels) and fetch levels from there if specified in the protocol link.