The Gotham Gazette (GG) is a good, wonky read especially if you're trying to make sense out of NYC politics. The article excerpted below might even help you understand what all those volunteer petitioners are doing out on the streets of NYC.

Reading GG, I learned a thing or two.

The election laws we now live by were developed in the 19th century to make the process more democratic. Previously the system was controlled by party
leaders who had complete control over which names appeared on the ballot.

Now thanks to those laws, there are petitioning requirements. According to GG, the purpose of these requirement is "to ensure that only those candidates with
huge campaign war chests or party backing have the wherewithal to get
on to the ballot — and stay there. Here's an excerpt from a piece called Understanding the Labyrinth: New York's Ballot Access Laws by DeNora Getachew and Andrea Senteno:

In order to get on the New York City primary election ballot this
year, candidates could begin collecting signatures for their
designating petitions on June 9 — 37 days[DMG3] before the last day to
turn in designating petitions for the primary election

The law is very specific about how many signatures candidates must
collect. They have to get 5 percent of the enrolled voters of the
political party in the political unit covered by the office — council
district, borough or the entire city — or the specific numbers
enumerated in the state's Election Law,
whichever is less. For a candidate for City Council, that number is 900
signatures, but as a cushion against petition challenges, the rule of
thumb is to obtain at least three times the legal minimum.

The candidates also must figure out is who is eligible to sign the
petitions and who can collect the signatures. While only registered
voters who are members of the candidate's political party and reside in
the district in question can sign the petition, any registered voter
who is a member of the candidate's political party and lives in New
York City can collect signatures. Voters are allowed to sign just one
petition per office.

The candidate has approximately five weeks to collect all of the
requisite signatures and file his or her designating petitions with the
main city Board of Elections
office between July 13 and 16, which complies with the state law
requirement that designating petitions be filed between the tenth
Monday and the ninth Thursday preceding the primary election.

That done, the challenge portion of the petitioning process begins.
According to the board's rules, it conducts a prima facie "review [of]
each cover sheet and petition to ensure compliance with the New York
State Election Law." This marks the first round of challenges to the
candidate's petition — but definitely not the last. The law allows any
voter registered who can vote for the candidate to file written
objections with the Board of Elections challenging that candidate's
designating petitions. Those challenges must be made within three days
of the filing of the petitions. [DMG4]

Once challenges are filed, the board holds hearings to assess the
validity of the challenges and issues a determination. In order to
appeal the board's decision a person must commence an action in state
Supreme Court – the lowest level court in New York's court system
"within 14 [DMG5]days after the last day to file a petition or within
three business days after the board makes a determination regarding the
invalidity of such petitions, whichever is later."

If the appeal involves a determination about whether a candidate's
name will appear on the ballot or a voting machine, the Supreme Court,
if possible, is supposed to issue a final order at least five weeks
before the day of the election. Candidates can appeal such decisions.