Jay-Z is a part owner of the Nets and he wants to move them to Brooklyn. Branch Rickey was the white general manager of the Dodgers who signed Jackie Robinson (referenced earlier). So if Jay-Z brought the Nets to Brooklyn, this would be groundbreaking because he’s a black NBA owner and he’d be bringing a new team to Brooklyn

Jackie Robinson was the Brooklyn Dodger player who broke the color barrier; “Dodger” means criminal — the “Artful Dodger” was Oliver Twist’s pickpocket mentor in Dickens' classic.

Jackie Robinson, as mentioned above, broke the color barrier in baseball. He is regarded as being one of the greatest to ever play the game. His high speed made him a great candidate to steal bases and get extra bases on his hits. Hence, “running base.”

Jay-Z uses word play about stealing and sinning to come up with how he “is” Jackie Robinson. Except when he runs base (sells drugs like freebase cocaine vs. Jackie running around the bases), he dodges jail time.

When Jay-Z raps along to the bass of the beat, he freestyles rather than writing raps beforehand (dodging the pen and/or Microsoft Word)

Another meaning is that while Jay-Z dodges the pen (short for penitentiary or prison, Jackie Robinson isn’t. However, the pen Robinson isn’t dodging likely is the one that was used to sign lucrative baseball contracts.

Another use for “dodge the pen” could be Jay-Z referencing the bullpen in baseball stadiums. The bullpen or “the pen” is where relief pitchers warmup before they come into the game. In order for a baseball player to get on base, they might need to get a hit off of a relief pitcher from the pen if the at-bat is later in the game (7th, 8th, 9th inning). However, relief pitching was less prominent in Jackie’s day than it is today.

Also note the use of the word “dodge,” when Jackie played for the Brooklyn Dodgers. The name of the Dodgers (now in Los Angeles) was derived from Brooklyn’s trollies which ran along the streets, and the city’s citizens had to “dodge” out of the way (Source).

For those keeping count: triple entendre

Jay-Z’s father left him when he was 11 and since his mom was always busy working, he started only hanging out on the streets and getting into criminal activity. More about the numbers game here

Brownsville is a neighborhood in Brooklyn and Bainbridge is a street in Brooklyn. He covers the whole borough

Buju Banton is a controversial Jamaican reggae singer. This whole opening is in a reggae style of voice

If a man questions my BK credentials (Bed-Stuy), he’s a goner

Also, if a man tests my “sty” as short for “style.” I.E. calls me out for anything

“Sty” could also be short for pigsty, which could be interpreted as a relatively dirty place (which would make sense because he’s talking about how rough Brooklyn is) or as a shelter where pigs are kept (which would simply be stating that Brooklyn is his home like a pigsty might be a pig’s home).

“Boom bye bye” is a Buju song about killing homosexuals (with lyrics such as “Boom bye bye Inna batty bwoy head” where “Batty boy” is a derogatory term for homosexual)

Hov has been talking about looking him in the eye in the previous lines saying thats not a problem but if you test his neighbourhood it will be.

Rappers often talk about how their raps are so great that, when you hear them, you have to rewind to listen to them again (see Eminem’s verse in “Forever” for another rewind brag) or because you simply didn’t understand all the complex lyricism on the first listen.