Why the highly formalized language of "Now comes..."?
"NOW COMES" is traditional ("Comes Now" is actually more common even though it is even more formal and awkward), a bit like "WHEREAS" in contracts.
Modern legal writing disfavors this wording in the first sentence of a legal document and I usually omit it unless I know that the judge is very old fashioned.
These days, when a lawyer is in front of a court in person, the lawyers starts to speak about something by saying "May it please the court" (another traditional phase emphasizing deference to the fact that the judge can throw you in jail if you are rude without a trial in a courtroom).
But, people used to say, "NOW COMES" instead and that phrase stuck in written form.
Why the seemingly random capitalization?
The capitalization is not completely correct in your example. Some people have the bad habit of capitalizing every word that they think is important, which is not proper in English. (For what it is worth, in German, all nouns are capitalized.)
Petitioner should be capitalized because it is being used as a proper noun in lieu of someone's name.
Court should be capitalized here because the rule is that the word Court is capitalized when you are talking about the court that you are in, but in lower case when you are talking about another court's rulings.
Enter was improperly capitalized.
Petitions is improperly capitalized.
(In general, probably as a residual of the fact that English is a Germanic language, verbs are almost never capitalized unless they are defined terms or are the first word in a sentence.)
Order is capitalized when it refers to a particular order that already exists, but should be in lower case here when it is referring to an order that is being requested in the future.