Open source is when the source code is known, closed source is when it's hidden. Open source software can be compiled by yourself, knowing that the code it's running from is safe (if you or other devs have inspected it beforehand). In closed source you have to completely trust the dev that the software he is providing is not malware or doesnt have a backdoor in it.

A closed source wallet software is a horrible idea. It could either have a huge bug in it that is hard to detect by their small dev team, instead of publishing it so that everyone can analyze it. Or the devs could be scammers and insert a backdoor in it, that will steal all your coins later.