What emulator did you download that tells you that message?
I do believe emulators are legal but the part that makes it illegal is the BIOS files, aka the files that actually make it operate like the real console. You are supposed to dump those files yourself using your own hardware. All PS1 emulators require you to obtain BIOS files before they will work properly, which is what I believe that message in your OP is referencing.
Same with ROMs, it is illegal to download or distribute ROMs, the way you are supposed to obtain them is by dumping them from real cartridges/discs. This is why no big companies have managed to strike down any emulators for copyright although I'm very sure they would love to. They aren't breaking any laws.
Now... are you going to get arrested or fined for downloading ROMs or BIOS files? Well, I think the police would rather go after the people who hosted the files first, before they go after you, unless you are bulk downloading a billion ROMs for some reason. Your ISP might catch you by monitoring your browsing history though. So you are
probably safe to get some of the rarer games... but I don't recommend breaking the law.