Press F4 on the title screen of any of the official games (some fanmade games might also allow this) to dump all the levels as LVL files. They'll be a "/dump/[game name]/" subfolder of wherever your NeoLemmix EXE is.
If you didn't mean the actual official levels, but just levels from older Lemmix format, you should be able to open these directly in the NeoLemmix Editor. If it identifies the graphic set incorrectly (rare, but it does happen), just change it to the correct one and it should work.