The source of the quote is Eliot's Shakespeare and the Stoicism of Seneca, an address to the Shakespeare Association in 1927 that was reprinted in the collection Selected Essays (1917-1932) (Faber and Faber, 1932).
The address begins by discussing various studies of Shakespeare's work by critics such as John Middleton Murry, Wyndham Lewis (The Lion and the Fox: The Role of the Hero in the Plays of Shakespeare, 1927) and Lytton Strachey. Some studies or essays tried to identify influences on Shakespeare:
We have had Shakespeare explained by a variety of influences. He is explained by Montaigne, and by Machiavelli. I imagine that Mr. Strachey would explain Shakespeare by Montaigne, though this would also be Mr. Strachey's Montaigne (for all of Mr. Strachey's favourite figures have a strong Strachey physiognomy) and not Mr. Robertson's. I think Mr. Lewis, in the intensely interesting book mentioned, has done a real service in calling attention to the importance of Machiavelli in Elizabethan England, though this Machiavelli be only the Machiavelli of the Contre-Machiavel, and not in the least the real Machiavelli, a person whom Elizabethan England was as incapable of understanding as Georgian England, or any England, is.
(Machiavelli's The Prince was banned in Elizabethan England but Elizabethans knew the gist of his works, or at least what they thought was the gist of his works, from other sources.)
Eliot does not believe that Shakespeare was "influenced" by Machiavelli's works; instead, he says,
I think that Shakespeare, and other dramatists, used the popular Machiavellian idea, for stage purposes; but this idea was not more like Machiavelli, who was an Italian and a Roman Christian, than Mr. Shaw's idea of Nietzsche—whatever that is—is like the real Nietzsche.
Eliot than makes the following suggestion:
I propose a Shakespeare under the influence of the stoicism of Seneca. But I do not believe that Shakespeare was under the influence of Seneca. I propose it largely because I believe that after the Montaigne Shakespeare (not that Montaigne had any philosophy whatever) and after the Machiavelli Shakespeare, a stoical or Senecan Shakespeare is almost certain to be produced.
Eliot believes that Shakespeare may have read some of Seneca's tragedies at school but not his prose, so if Shakespeare was influenced by Seneca, it was probably through memories from school and the works of George Peele (1556 – 1596) and Thomas Kyd. Eliot then goes on to discuss how Shakespeare's play display a new attitude that is "derived from Seneca" even though it is "not the attitude of Seneca", namely "the attitude of self-dramatization assumed by some of Shakespeare's heroes at moments of tragic intensity".
Eliot then expresses disagreement with critics such as Wyndham Lewis who see Shakespeare as representing a specific philosophy:
I cannot see in Shakespeare either a deliberate scepticism, as of Montaigne, or a deliberate cynicism, as of Machiavelli, or a deliberate resignation, as of Seneca. I can see that he used all of these things, for dramatic ends: you get perhaps more Montaigne in Hamlet, and more Machiavelli in Othello, and more Seneca in Lear.
Lewis sees Shakespeare and Chapman as dramatists who think. Eliot disagrees:
The poet who 'thinks' is merely the poet who can express the emotional equivalent of thought. But he is not necessarily interested in the thought itself.
Wyndham Lewis wrote, "We posses a great deal of evidence as to what Shakespeare thought of military glory and martial events." Eliot responds (emphasis mine),
Do we? Or rather, did Shakespeare think anything at all? He was occupied with turning human actions into poetry.
I would suggest that none of the plays of Shakespeare has a 'meaning', although it would be equally false to say that a play of Shakespeare is meaningless. All great poetry gives the illusion of a view of life.
He then goes on to discuss the example of Dante:
[We think] Dante has a 'philosophy', therefore every poet as great as Dante has a philosophy too. Dante had behind him the system of St. Thomas, to which his poem corresponds point by point. Therefore Shakespeare had behind him Seneca, or Montaigne, or Machiavelli; (...). I can see no reason for believing that either Dante or Shakespeare did any thinking on his own. The people who think that Shakespeare thought, are always people who are not engaged in writing poetry, but who are engaged in thinking, and we all like to think that great men were like ourselves.
Summary: "none of the plays of Shakespeare has a 'meaning'" means that Shakespeare's plays are not the expression of philosophical ideas, since Shakespeare was a poet, not a "thinker".