What is the significance of the two intrigues by Mirabell and Mr. Fainall? How are they linked with the final scene of the play?

The Way of the World by William Congreve involves a series of marriages, marriage proposals, and extramarital affairs. As Mr. Fainall says in Act III, Scene 18,

Why, then, Foible’s a bawd, an errant, rank match-making bawd. And I, it seems, am a husband, a rank husband, and my wife a very errant, rank wife,—all in the way of the world.

At this point in the play Mrs. Marwood is having an affair with Mr. Fainall. She overhears a conversation that reveals that Mrs. Fainall is having an affair with Mirabell, and she passes this information on to Mr. Fainall. This quote suggests that these extramarital affairs are commonplace in eighteenth-century British society; Congreve draws his title from this idea.
Both Mr. Fainall and Mirabell have secret plots, or intrigues, that they use to their advantage in the play. Mirabell makes a plan to have his servant, who pretends to be his wealthy uncle, woo Mrs. Wishfort. After she agrees to marry the false uncle, Mirabell plans to reveal his true identity as a mere servant. Then, Mirabell can offer to help Lady Wishfort out of this shameful situation if she will allow Mirabell to marry Millamant (who he loves at this point in the play). Lady Wishfort's approval of their marriage is critical since she controls part of Millamant's fortune.
Mr. Fainall then begins his own plan (or intrigue). He finds out the true identity of Lady Wishfort's fiance, a household servant. He exposes this information to Lady Wishfort and then demands money from her to cover up her shameful situation. He demands the remainder of his wife's fortune that Lady Wishfort controls; he also wants Millamant's money that she has refused by choosing to marry someone her aunt does not consent to. In Act IV, Scene 10, he reminds Lady Wishfort:

FAINALL: . . . But I’m prepared for you, and insist upon my first proposal. You shall submit your own estate to my management, and absolutely make over my wife’s to my sole use, as pursuant to the purport and tenor of this other covenant. . . . This, my Lady Wishfort, must be subscribed, or your darling daughter’s turned adrift, like a leaky hulk to sink or swim, as she and the current of this lewd town can agree.
LADY WISHFORT: Is there no means, no remedy, to stop my ruin? Ungrateful wretch! Dost thou not owe thy being, thy subsistence, to my daughter’s fortune?
FAINALL: I’ll answer you when I have the rest of it in my possession.

Fainall insists that he will have this money; this would essentially take the majority of Lady Wishfort's fortune away from her. Lady Wishfort asks if anyone will come to rescue her out of her horrible predicament. Mirabell then offers to help her:

MIRABELL: . . . I am resolved I’ll serve you; you shall not be wronged in this savage manner.
LADY WISHFORT: How? Dear Mr. Mirabell, can you be so generous at last? But it is not possible. Harkee, I’ll break my nephew’s match; you shall have my niece yet, and all her fortune, if you can but save me from this imminent danger.
MIRABELL: Will you? I take you at your word. I ask no more. I must have leave for two criminals to appear.

In return, Lady Wishfort agrees that Mirabell can marry his niece (Millamant). Mirabell then brings in two people: Foible and Mincing (two servants). These two women prove that Mr. Fainall was having an affair with Mrs. Marwood. Mr. Fainall is not greatly upset by this revelation, since it could still bring shame upon Mrs. Fainall unless Lady Wishfort gives him money to quiet him.
Finally, Mirabell reveals his most significant secret. After Mirabell's affair with Mrs. Fainall, he helped her find a husband (Mr. Fainall). (This would prevent any disgrace if she were pregnant.) Mirabell and Mrs. Fainall did not trust Mr. Fainall (though she did agree to marry him), so they made a plan to protect her money by appointing Mirabell as manager of her fortune:

MIRABELL: . . . I say that this lady, while a widow, having, it seems, received some cautions respecting your inconstancy and tyranny of temper, which from her own partial opinion and fondness of you she could never have suspected—she did, I say, by the wholesome advice of friends and of sages learned in the laws of this land, deliver this same as her act and deed to me in trust, and to the uses within mentioned. You may read if you please [holding out the parchment] . . .
FAINALL: Very likely, sir. What’s here? Damnation! [Reads] A Deed of Conveyance of the whole estate real of Arabella Languish, widow, in trust to Edward Mirabell. Confusion!
MIRABELL: Even so, sir: ’tis the way of the world, sir; of the widows of the world. I suppose this deed may bear an elder date than what you have obtained from your lady.

Ultimately, both Mirabell and Fainall were involved in extramarital affairs. Both used conniving and deceitful plots to get what they wanted. In the end, Mirabell triumphs over Fainall and gains both the management of Mrs. Fainall's fortunes and the freedom to marry Millamant with Lady Wishfort's approval. Mr. Fainall is tricked and manipulated by Mirabell yet again.
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