Here's a breakdown:

" Less dare I, to avoid self-glorification, encumber Your weighty occupations, Illustrious, High and Mighty, with the trouble, care, solicitude and continual zeal with which I have endeavored to promote the increase of population, agriculture and commerce; the flourishing condition whereunto they were brought, not through any wisdom of mine, but through God's special blessing,

Here, he's talking himself up about how he was able to turn the fortunes of the New Netherland colony around, after the disastrous administration of his predecessor, Willem Kieft. There had been a war with the Indians that came to be called Kieft's War, and was seen by the New Amsterdammers as Kieft's fault. He'd been hostile to the Indians, and didn't seem to negotiate in good faith, and New Amsterdam was plunged into war at Kieft's direction. A lot of Dutch colonists died. New immigrants had stopped coming, and the colony was on the brink of failure.

The turnaound was due much to Stuyvesant's leadership--although he did some questionable things as well--but in that passage, he's trying to be humble by saying it wasn't his wisdom that had helped the colony thrive in recent years, but it was a result of "God's special blessing".

and which might have been more flourishing if your formerly dutiful, but now afflicted, inhabitants of that conquest had been, Illustrious, High and Mighty, protected and remained protected by a suitable garrison, as necessity demanded, against the deplorable and tragical massacres by the barbarians, whereby (in addition to ten private murders) we were plunged three times into perilous wars, through want of sufficient garrisons

Here, Stuyvesant is talking about his requests for additional defenses for New Amsterdam, which the Dutch West India Company never followed through on. He blamed this lack of military might for the recent wars with the local Indian communities (who he refers to as "the barbarians", not an uncommon term by which they were called by the Dutch at the time--they were also often called "the Turks"). The three "perilous wars" he is referring to were likely the Peach Tree War (1655), the First Esopus War (1659), and the Second Esopus War (1663).

especially had they, on the supplicatory remonstrances of the people and our own so iterated entreaties, which must be considered almost innumerable, been helped with the long sought for settlement of the boundary

Here, Stuyvesant is complaining about how the Dutch authorities had never settled the dispute with the English over control of Long Island and the Hudson River Valley. The Dutch controlled what's now Brooklyn, Queens, and waht's approximately Nassau County, while the eastern end of Long Island that's now Suffolk County was controlled by the English, or thereabouts. The border between the Dutch and English there had never been properly settled, and the land upstate east of the Hudson, where New York now borders Connecticut, had also never been formally resolved, and there had been land disputes between the Dutch and English up there as well. So Stuyvesant is saying that he and the New Netherlanders had made "innumerable" pleas to the Dutch West India Company to get these issues resolved, but it had never been done, and the result was now the loss of the entire colony to the English.

or in default thereof had they been seconded with the oft besought reinforcement of men and ships against the continual troubles, threats, encroachments and invasions of the English neighbors and government of Hartford Colony, our too powerful enemies.

This is more of the same. In lieu of a treaty settling the land dispute between the Dutch and the English, the Dutch had never sent any kind of military force to help keep the Dutch land in Dutch control, and now their "too powerful enemies" have been able to take control of the whole of the New Netherland colony, including New Amsterdam.

To read up on the history of New York during the Dutch period, I recommend reading The Island At The Center of the World by Russell Shorto. It's well-sourced, but it's also written in a narrative style, so it's a relatively easy read. Another alternative is The Colony of New Netherland: A Dutch Settlement in Seventeenth-century America by Jaap Jacobs, which covers much of the same material. Both cover Stuyvesant's surrender and the circumstances surrounding it.

Stuyvesant wanted to fight, but the city council lobbied against it, but Stuyvesant wouldn't listen. He went into the fort (located where the American Indian Museum now stands, next to Battery Park) and two church ministers were sent to plead with him to not try to fight the English, but Stuyvesant was still unmoved. A town meeting was held after the English sent over their surrender terms--which were pretty generous. All they requested was control of the fort, but otherwise, they promised things would stay the same. This turned out to be a bit of an exaggeration, but in any case, Stuyvesant either didn't believe them, or believed that the townspeople wouldn't side with him if he knew about them, or both, so he tore up the letter that had been delivered to him from the English. It didn't much matter, because few of the Dutch population were willing to fight. Most of the population had been through three Indian wars, and other smaller violent events, and saw the English Navy's power as too much to overcome. If the Indians could destroy much of the town, the English were going to obliterate it. As a result, 93 townspeople of New Amsterdam signed a petition urging Stuyvesant to surrender--including his own son Balthasar. If Stuyvesant didn't surrender, he'd find he would have few soldiers willing to fight behind him.

At that point, Stuyvesant agreed to open up negotiations, and got in writing from the English a lot of guarantees that the English had promised in general terms: they could keep their Dutch church and policy of religious tolerance, the lower level government officials could keep their offices (though the English would take over the ultimate administration), they could keep their individual land claims, and so on.

As it turned out, the English didn't entirely keep their word: the Dutch churches were allowed to stay open, but all the Dutch citizens were almost immediately forced to pay a tax to the English church--a church with no buildings, which had next to no ministers, and which none of the citizens belonged to. The Dutch were allowed to keep their organization of government, but the official language of the government was changed to English, which few of the citizens understood, so when they tried to bring a lawsuit in criminal or civil court, they were at a disadvantage. After a while, they were also denied the right to vote or to hold office unless they swore an oath to the King of England. For all these reasons, and more, there ended up being hostility on both sides, culminating in the Dutch taking back control of the colony twice. The first was in 1673-74 during the Third Anglo-Dutch War, when the Dutch Navy showed up and pulled the same stunt the English Navy had pulled on Stuyvesant. The treaty at the end of that war gave the colony back to the English. But a more direct result of all the violations of the surrender agreement was Leisler's Rebellion in 1689, after the Glorious Revolution back in England that had recently resulted in a similar citizen uprising in New England. The Dutch farmers under Jacob Leisler took control of the colony for eighteen months before the English Navy bothered to show up, and then negotiated for an end to the rebellion. Leisler and his son-in-law were executed, despite the fact that the initial rebellion had been carried out without a shot fired, and next to no shots were fired when the English Navy showed up, so nobody had been killed. For the next 20-30 years, New York politics was organized around "Leislerian" and "anti-Leislerian" political factions/parties in the New York Assembly, but by the 1730s, the English/British outnumbered the Dutch in significant enough numbers that the Dutch peasantry alone wasn't enough to hold onto power, and the political factions began to reorganize around the emerging demographics of the colony.

For a history of the post-Dutch period, a couple of good sources are Colonial New York: A History by Michael Kammen, originally published in 1975 and part of a larger series about the colonial history of each of the Thirteen Colonies. More recent and probably more readable is the Pulitzer Prize-winning Gotham: A History of New York City to 1898 written by Edwin G. Burrows and Mike Wallace.

EDIT: I see you edited your post, but I thought you might be interested in this answer anyway.