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The qualifications applied to any man who:

Was 21 years or older
Was a resident of a town in Massachussets for at least one year
Owned an estate in that town that produced three pounds of income OR owned in any estate worth sixty pounds in said town

Somewhere between 80 and 90 percent of adult men met these legal requirements to vote. However, as /u/RTarcher says, this was even higher in practice due to the practice of reinterpreting the legal requirement to "any income" (an agricultural laborer could expect at least four pounds of annual income on the low end) and, because towns were responsible for choosing their representation, often let even that minimal requirement slide in the interest of the tight-knit community.

As for your question about the principle/theory behind the qualifications for representation:

The majority (but not quite dominant) theory in the colonial and early US was that only those who owned property had sufficient "discretion" to vote. "Discretion" was a common term used and concept discussed in political literature of the time. It's the ability of a voter to make decisions free of excessive influence from whoever they were dependent on. Opinions on who had sufficient and insufficient discretion varied widely, focused on these classes of person:

Those under the age of majority. Universally considered not to have sufficient discretion, but the precise age of majority was argued (generally either 18 or 21)
Landowning white men. Of age (by default - otherwise they couldn't own property or sign contracts) and universally thought to be independent-enough of outside influence to vote with reason.
Women. Arguments generally were that women did have sufficient capacity to vote, but that they didn't have discretion because they were economically and socially dominated by husbands and/or because feminine qualities were incompatible with voting. (The Essex Result: "women what age soever they are [do not have sufficient discretion] not from a deficiency in their mental powers, but from their natural tenderness and delicacy of their minds, their retired mode of life, and various domestic duties")
Free, land-owning black men. Considered to have discretion due to their legal equality, or not due to racist beliefs that they lacked the decision-making capacity of whites. The 1780 MA constitution makes no mention of this class, and likely did extend voting rights to those who otherwise qualified.
Slaves. Universally considered not to have discretion, including by political theorists otherwise against slavery. Since slaves were both almost universally uneducated and universally under compulsion by their masters, slaves would be compelled to vote for whoever their overseers commanded.
Non land-owning whites/the poor/"everyone else." By far the most contentiously discussed class, and the one most relevant to your question. Arguments against the discretion of this class emphasized their lack of "stake" in the well-being of the community, that they would exercise the tyranny of the mob to vote the property away from the wealthy, that they were generally less-educated, and a huge amount of endemic classist loathing directed at the poor (commonly described as the "profligate and idle," "waste persons" - the origin of the term "white trash"), and even "the excreta of society." Political elites of the era, even such luminaries of equality like Ben Franklin, often fucking despised the landless poor. This was tied to beliefs in an agricultural ideal; cultivating land was thought to cultivate men - marginal land produced marginal men, and landless men were rootless wastrels (though landlords were cultivators by working "through" their tenants). Arguments for the discretion of the poor were that these were people bound by the law and so should have a say in it, that land didn't translate to any special wisdom, that the state was one of the few recourses the poor had against the power of the wealthy, and ideological appeals to liberty.

The Massachusetts legislature had proposed a constitution that offered universal male suffrage in 1778, but it was rejected by voters for (among a large list of reasons) not including a property qualification. The influential The Essex Papers argued not only that a legislature representing all persons would be unwieldy in size but would hand power to the poor whites without discretion. The town of Sutton rejected it based on this provision, which closely matches the reasoning of other rejections:

Provided that Men who lie a Burden upon the publick or have little or no Property ought not to have a full or equal Vote with those that have an estate, in voting for a Representative, for then a Representative may be chosen without any property, and as the proposed Constitution stands there is all the chance that could be wished, for designing mischevious. Men to purchase themselves seats in the House; for poor, shiftless spendthrifty men and inconsiderate youngsters that have no property are cheap bought (that is) their votes easily procured Choose a Representative to go to court, to vote away the Money of those that have Estates"

Adam's draft of the 1780 constitution included a property requirement, but the legislature went further, substituting the word "resident" for "persons" explicitly to conform to the text of a law allowing the expulsion of poor people.

This wasn't universal across the state, however. The town conventions of Richmond, Dorchester, Mansfield, and Lenox (among others) all objected to this qualification on the basis that it was essentially criminalization of poverty and an infringement on their rights. From Lenox's response:

We conceive this Article declares Honest Poverty a Crime for which a large Number of the true and faithfull Subjects of the State, who perhaps have fought and bled in their Countrys Cause are deprived of the above mentioned Rights (which is Tyranny) for how can a Man be said to [be] free and independent, enjoying and defending Life and Liberty and protecting property, when he has not a voice allowed him in the choice of the most important officers in the Legislature, which can make laws to bind him

Their objections were not the prevailing sentiment, however, and the 1780 constitution was ratified with property qualifications.

To answer your follow-up about Adams' personal reasons for including a property requirement for voting, he articulated a version of the discretion argument along with his belief that political power was tied to property. He argued in an open letter that there had to be some line drawn, as there was one drawn for an age of majority:

What Reason Should there be, for excluding a Man of Twenty years, Eleven Months and twenty-seven days old, from a Vote when you admit one, who is twenty one? The Reason is, you must fix upon Some Period in Life, when the Understanding and Will of Men in general is fit to be trusted by the Public. Will not the Same Reason justify the State in fixing upon Some certain Quantity of Property, as a Qualification.

He further warned that, though such men were certainly capable of voting with reason and were affected by laws, this alone wasn't sufficient for necessary discretion and would start a slippery slope:

The Same Reasoning, which will induce you to admit all Men, who have no Property, to vote, with those who have, for those Laws, which affect the Person will prove that you ought to admit Women and Children: for generally Speaking, Women and Children, have as good Judgment, and as independent Minds as those Men who are wholly destitute of Property: these last being to all Intents and Purposes as much dependent upon others, who will please to feed, cloath, and employ them, as Women are upon their Husbands, or Children on their Parents

and his (typical of many political elite at the time) obsessive fear of threats to what he believed was the necessary class structure:

every Man, who has not a Farthing, will demand an equal Voice with any other in all Acts of State. It tends to confound and destroy all Distinctions, and prostrate all Ranks, to one common Levell.

As you might expect from his openness to the cogent political reasoning abilities of the poor and even women, Adams wasn't for the oppression (political or otherwise) of the indigent as a punishment for their perceived idleness and sloth. He advocated that policy should alleviate poverty and explicitly spread suffrage by making land ownership affordable and that regulation should enable even small estate owners to keep their land. In any event, this ideal wouldn't have much time to be tried out before urbanization and industrialization increased the landless working/middle class as agriculture proportionally declined. This new sociopolitical class demanded updated suffrage laws to recognize them de jure, which were ratified in the early 1800s.