Frederick Temple (1821-1902), Archbishop of Canterbury, who co-wrote the Anglican response to Pope Leo XIII, on this matter
Capsule summary: the Anglicans broke the line of apostolic succession in the 16th century. This was necessary for valid ordination of priests, without which there is no valid consecration and Sacrifice of the Mass and Holy Eucharist.
The October 1992 edition of This Rock gave a relatively short explanation:
Q: An Anglican priest tells me that his holy orders are valid and that he can consecrate the Eucharist and grant absolution. I've heard the opposite is true. What is the Catholic position?
A: Although Catholics and many traditional Anglicans are now enjoying an era of unprecedented friendliness and increased mutual cooperation, there still remains the touchy subject of whether Anglican holy orders are valid. The Catholic Church continues to regard them as invalid.
In 1896 Pope Leo XIII issued his apostolic letter Apostolicae Curae, in which he upheld the Church's position that Anglican orders are "absolutely null and void." When the first Anglican Archbishop of Canterbury, Thomas Cranmer, came to power under King Henry VIII, he drastically modified the rite of ordination, eliminating all references to a sacrificial priesthood.
Since to be valid the sacraments must have the proper form and matter, grave questions immediately arose as to the validity of Anglicanism's new form of holy orders. Upon further study, the Catholic Church determined that, although an ordination might be attempted by a valid though heretical Catholic bishop, because the Anglican rite of ordination had been so distorted it could no longer effect a valid ordination.
Thus, within a generation or two after the inception of the Anglican Church there were no validly consecrated Anglican bishops (the original Catholic bishops who had gone into heresy having since died). Therefore the Anglican bishops (who technically weren't bishops at all nor even priests) couldn't validly ordain men to the priesthood.
There is, though, a further complication. Some candidates for the Anglican priesthood, recognizing the sterile nature of their church's holy orders, have received ordination at the hands of validly ordained schismatic bishops (such as the Old Catholics, who broke from Rome in the nineteenth century). Assuming these bishops used the proper rite and had the necessary intention, those ordinations would be valid, though illicit. The problem is that it's extraordinarily difficult to ascertain whether an individual Anglican priest's orders are valid or not.
That's why Anglican priests who wish to become Catholic and function as priests must be ordained anew in the Catholic Church. They are always ordained "absolutely," not "conditionally"--that is, the working presumption for all of them is that they were not validly ordained while in the Anglican Church, no matter who their ordaining bishops were.
Of course, Anglicans disagree and there has been a huge dispute about the whole thing.
The key document is the papal encyclical Apostolicae Curae ("On the Nullity of Anglican Orders"), dated September 18, 1896, by Pope Leo XIII.
Here is an article on it: Leo XIII's Decision on Anglican Orders: The Extrinsic Argument (Paul R. Rust)
In his Doctrinal Commentary on the Concluding Formula of the Professio fidei (29 June 1998), on Pope John Paul II's Apostolic Letter Motu Proprio, Ad Tuendam Fidem (18 May 1998), Pope Benedict XVI (then Joseph Cardinal Ratzinger) reaffirmed the invalidity of Anglican orders:
With regard to those truths connected to revelation by historical necessity and which are to be held definitively, but are not able to be declared as divinely revealed, the following examples can be given: the legitimacy of the election of the Supreme Pontiff or of the celebration of an ecumenical council, the canonizations of saints (dogmatic facts), the declaration of Pope Leo XIII in the apostolic letter Apostolicae Curae on the invalidity of Anglican ordinations....
Fr. Brian W. Harrison, a very insightful writer, has written the article: "Recent Thought on Anglican Orders" (scroll down a page or two).
Fr. Christoper Phillips has written: "Anglican-Catholic Relations: The Quest For Unity."
"The Trouble With Anglo-Catholicism," Robert Ian Williams, This Rock, Sep. 2001.
Catholic Encyclopedia (1913): "Anglican Orders"
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For the Anglican response, see the 1897 document, Saepius Officio.
Holy Orders and Apostolic Succession (AskThePriest.org)