Why do Elinor and Edward act with such restraint toward each other?

While it is correct to say that both Elinor and Edward are calm and decorous in their expressions of esteem and fondness and that their level of emotional display stands in contrast to the heightened display of Marianne and Willoughby, it is not correct to suggest that Elinor's reason is modesty and that Edward's reason--that Edward's sole reason--is his secret engagement. In regard to his secret engagement to Lucy, it is a error to suggest that Edward's mother, Mrs. Ferrars, approves of or favors Lucy, whom she has never met and of whose engagement to Edward she has no idea.
Elinor
What is true to say is that Elinor governs her emotions and her expectations through sense—half of the principle theme of the novel—that is borne of a strong, self-confident, capable personality and social interactions, which are not based in modesty (to have modesty: to have a moderate or humble estimate of one's worth and importance to self and society) but in a sound and reasonable psychological nature in which sense is embedded. When considering sense versus sensibility and the differences or similarities between Mrs. Dashwood, Elinor, and Marianne, one must note Austen's foray into the discussion now known as "nurture versus nature": while Elinor and Marianne are raised in the same environment by the same persons, they have exceedingly different personalities and expressions of sense and sensibility.
Edward
What is true to say about Edward is that he is restrained in his display of emotion for three reasons. The first, of course, is his secret engagement, but the second and third are that it is Edward who is modest, as revealed by the narrator who conveys Elinor's thoughts, "... charity with the modesty and worth of the other [Edward compared to Robert] (Ch 36)," and that he received an inferior private education rather than public education; he received "the misfortune of a private education (Ch 36)."
Edward's education was, by his own and Robert's accounts, inferior in that it was "private" rather than "public." It left Edward without the depth of education that would rendered him skilled and without practice in the social interactions that might compensate for his natural modesty and diffidence. He might have still preferred the quiet life of the clergy, yet he would not have been so inexperienced with society as he is had he had a public education.
English Public and Private Education
It is important to understand the difference between public and private English education. English public education is for the elite and wealthy (or for those on scholarship, which is how Geoffrey Chaucer, Edmund Spenser, and William Shakespeare were educated: at public school on scholarship) and is conducted for "public" groups at boarding schools, such as Eaton, which England's Princes William and Henry attended.
English private education is the placement of a student with a master or tutor, often a clergyman or some other scholar, in their private residence—sometimes with a small group of other private students and sometimes as the sole student—for an entire education or for a specialized part of an education. English scholar, Christian apologist, and novelist C. S. Lewis discusses his private education for maths in his autobiography Surprised by Joy. He was struggling with maths, so he was placed with a private tutor for the year preceding his entrance exams for Oxford.
The disadvantages of private and public education (each of which has its individual strengths as well) are seen in the experiences of Robert and Edward: a private education can foster natural tendencies to reclusiveness and false modesty, as with Edward, while a public education can foster proclivities toward arrogance and false pride, as with Robert. This idea of nurturing through education is, of course, one of the themes which is explored by Austen in Sense and Sensibility and which was advanced by John Locke (1632-1734) in the 1700s (Austen, 1775-1817).
Edward, Lucy, and Mrs. Ferrars
The premise by which Lucy introduces the interesting topic of Edward to Elinor is the question of whether Elinor can tell Lucy about the characteristics of Mrs. Ferrars, Edward's mother: "are you personally acquainted with your sister-in-law's mother, Mrs. Ferrars? ... perhaps, you cannot tell me what sort of a woman she is?" Elinor, in shock and surprise, says she didn't know Lucy was connected with the Ferrars. Lucy replies that she is not but that her hope is that she will soon be: "Mrs. Ferrars is certainly nothing to me at present—but the time MAY come—how soon it will come must depend upon herself—when we may be very intimately connected."
Elinor understands this veiled social reference to an engagement but supposes Lucy must mean Robert Ferrars: "'No,' replied Lucy, 'not to Mr. ROBERT Ferrars ... but,' fixing her eyes upon Elinor, 'to his eldest brother.'" When Elinor presses the point by saying that she has never heard Edward "even mention" her name, Lucy responds by saying that the engagement "was always meant to be a great secret" and that their secret engagement must never "reach his mother; for she would never approve of it," because Lucy will "have no fortune," and Mrs. Ferrars "is an exceedingly proud woman."
Thus it is clear that it is quite an error to say that Edward is restrained in his emotional expressions because of a secret engagement that his mother favors. There is a potential engagement to a Miss Morton, "the Hon. Miss Morton, only daughter of the late Lord Morton, with thirty thousand pounds," that Mrs. Ferrars favors and, indeed, advances, although to no avail.
The reason Edward's secret engagement restrains his display of emotional feeling is a much more generic one that Austen explored to good effect in Emma; consider Knightley's accusations aimed at Churchill's behavior toward Emma. In Georgian society (of which the Regency period was a part), attentions given by an eligible young man to a marriageable young woman were expected to result in a proposal of marriage, as was illustrated so clearly by Austen in Pride and Prejudice; consider the community's and the Bennets' expectations of marriage between Jane and Bingley. It was dishonorable for eligible and marriageable persons to behave frivolously; it was dishonorable for a young man to raise expectations through his prolonged or exclusive attentions to a young woman.
Consequently, when a secret engagement exists, an interested person cannot know that eligibility does not exist; therefore unrestrained attentions will be understood as the prelude to a proposal of marriage. Edward was restrained toward Elinor because he knew that, being secretly engaged to Lucy--with or without Mrs. Ferrars' favor of Lucy and happily or unhappily engaged--he could never follow through with a proposal to Elinor.
Elinor's security sunk; but her self-command did not sink with it. "Four years you have been engaged," said she with a firm voice. "Yes; and heaven knows how much longer we may have to wait. Poor Edward! It puts him quite out of heart." Then taking a small miniature from her pocket, she added, "To prevent the possibility of mistake, be so good as to look at this face. It does not do him justice, to be sure, but yet I think you cannot be deceived as to the person it was drew for.—I have had it above these three years." She put it into her hands as she spoke; and when Elinor saw the painting, whatever other doubts her fear of a too hasty decision, or her wish of detecting falsehood might suffer to linger in her mind, she could have none of its being Edward's face. She returned it almost instantly, acknowledging the likeness. (Ch 22)
After Edward's and Lucy's engagement becomes a matter of public scandal, Colonel Brandon offers the Delaford living (living: clergical position and income awarded for life) to Edward so that he and Lucy might advance their intention of marrying. This plan of Edward taking the Delaford lining is motivated by two things. Firstly, Edward has always preferred the profession of clergyman above all options his family proffered. Secondly, having already taken a degree at Oxford, he only required a short "two or three months [to] complete his ordination" into the clergy.
[Brandon said,] "I understand that he intends to take orders. Will you be so good as to tell him that the living of Delaford, now just vacant, as I am informed by this day's post, is his, if he think it worth his acceptance." Elinor, "I know so little of these kind of forms, that I can hardly even conjecture as to the time, or the preparation necessary; but I suppose two or three months will complete his ordination."
In the happy twist of the resolution, the Delaford living and Edward's ordination will benefit Elinor and Edward instead of Lucy and Edward.