What is the meaning of joke "may it plizzle the cozizzle" in Futurama?

Is there some cultural context of this joke I am missing?

Yes.

Is it funny because "plizzle" rhymes with "cozizzle", and "cozizzle" mimics some words from the Snoop Doggs songs?

Yes as well. Snoop Dogg popularized the slang (in black culture in the US) of adding the -izzle suffix (and sometimes infix) to various words. He does it in many songs, including Drop It Like It's Hot and Tha Shiznit. He also does it as a guest in other rappers' songs, such as the intro to Dr. Dre's The Chronic.

The -izzle suffix, however, existed long before Snoop Dogg more or less made it famous. It was originally "used by African American pimps and jive hustlers of the 1970s," according to the Wiktionary article to which I will link.

Check out the Wiktionary article on the matter, and do a Google search for snoop dogg izzle to get lots of info.

While other answers have been correct in identifying Snoop Dogg's popularization of the term, I think some explanation on the actual joke may be of use to you as well.

As you already spotted, "If it plizzles the cozizzle" is meant as a Snoop Dogg-style translation of "If it pleases the court".

Keep in mind that Futurama is set in the future. The fact Leela uses that statement in an actual court, and no one even laughs or objects, points towards that being the correct thing to say in the year 3000.

If we hear Shakespearean English today, it sounds haughty. As the only people still occupied with the older versions of English (or traditional pronunciations) generally are. Its historical context (and contrast by using it today) also carries a connotation of intellectualism.
However, during Shakespeare's day, that was the lingua franca, i.e. how everybody spoke. It was normal, maybe even considered vulgar dialect (as the actual upper class at that point spoke Latin/French (not sure for English history, to be honest)).

According to Futurama's joke, the same has happened to the izzle-speak. It used to be a lingua franca/vulgar way of speaking to us, but over time, it was regarded as more historical, and in turn more intelligent.

If it stands to reason that past languages sound overly haughty to us, it must stand to reason that the people in the year 3000 regard our historical (to them) izzle-speak to sound haughty.