To illegalize something is to make it illegal, or to announce that it's against the law. When a city's police set a new curfew, they illegalize kids and teenagers being out at night.
The verb illegalize is a somewhat informal way to say "make something illegal." Your town might illegalize smoking in public places or your state senators could vote to illegalize dog fighting. In either case, something previously allowed is being made illegal. It's considered better grammar to say "make illegal," or even "criminalize," since illegalize is a relatively new verb constructed from illegal, "against the law," and the verb-forming suffix ize.