Ben Jonson's contemporaries seem not to have understood it either. And here is Ben Jonson's own explanation:
I remember, the Players have often mentioned it as an honour to Shakespeare, that in his writing, (whatsoever he penn'd) hee never blotted out line. My answer hath beene, would he had blotted a thousand. Which they thought a malevolent speech. I had not told posterity this, but for their ignorance, who choose that circumstance to commend their friend by, wherein he most faulted. And to justifie mine owne candor, (for I lov'd the man, and doe honour his memory (on this side Idolatry) as much as any.) Hee was (indeed) honest, and of an open, and free nature: had an excellent Phantsie; brave notions, and gentle expressions: wherein hee flow'd with that facility, that sometime it was necessary he should be stop'd: Sufflaminandus erat; as Augustus said of Haterius. His wit was in his owne power; would the rule of it had beene so too. Many times hee fell into those things, could not escape laughter: As when hee said in the person of Cæsar, one speaking to him; Cæsar thou dost me wrong. Hee replyed: Cæsar did never wrong, but with just cause: and such like; which were ridiculous. But hee redeemed his vices, with his vertues. There was ever more in him to be praysed, then to be pardoned.
Ben Jonson's point being, as this blog explains, that if he wrote as well as he did without revising his work, just think how much better he would have written if he had revised it.
I think there is a clue to this just a little farther in the passage itself, where the speaker goes on to mention his "state"... and I think it is in the notion of "statehood" that these expressions (and perhaps their understanding) are to be grounded.
That is, the speaker is speaking from the state of not just being a man, but man, as in ALL men, and further, ALL (of) "man", that is, a state of mind not just identifying with all the members of that given set, but to represent them singularly, collectively, and superlatively (cf. expressions like "I am woman, hear me roar!").
Additionally, there is an allusion to "man" being used here as an abstraction, not just a concrete representative of same, or all of them, but the whole idea of being a man (or woman, later), as in expressions like "I am VENGEANCE," or similar identifications with abstractions of concrete things, or abstract concepts themselves. This identification-with-the-concept is what gives the speaker the presumed mandate to speak on that entire concept's behalf, as he indeed goes on to do throughout the quoted passage.
However, as the somewhat more modern examples above suggest, I am not sure this is so much a feature of Shakespearean English, but rather, that Shakespeare's prose simply uses more of this forceful stuff. Indeed, I would bet you a shirt frill that if H.G. Wells dumped you in ye local publick house in Merry Ole Stratford-Upon-Avon, you'd be hard pressed to hear any of the locals express such prose while quaffing their brews.
But that, of course, is pure speculation on my part. :o)
Best Answer
I understand it to mean both of those things. Shakespeare managed to pack a lot of meaning into his words. He would have been aware of all the possible senses.
If you wish to answer the question for yourself, think about what the nature of heart is in metaphor.
If we set aside science and medicine, the word 'heart' is most often used in relation to love and to courage.
The following website archives many sayings about the heart.
Heart Quotes and Proverbs
http://heartquotes.com/Heart-quotes.html
ADDENDUM
The word 'courage' derives ultimately from the Latin 'cor' meaning 'heart'.