Why does Theseus decide to let Lysander and Demetrius get married on his own wedding day in act 4, scene 2?

The decision you refer to takes place in act 4, scene 1, which occurs on the day Theseus, Duke of Athens, is set to wed Hippolyta, Queen of the Amazons. In this scene, Theseus, Hippolyta, and Egeus discover the four lovers—Demetrius, Lysander, Hermia, and Helena—sleeping outdoors and soon find out that affections between them have shifted since the beginning of the play.
Early in the play, Egeus demands that his daughter, Hermia, marry Demetrius. Since she is unwilling and loves another man, he takes his daughter to court, where Duke Theseus lays down the Athenian law. Theseus orders that Hermia either marry Demetrius according to her father's will, choose a celibate life as a nun, or suffer the death penalty. To avoid those fates, Hermia sneaks away with her lover, Lysander. The jealous Demetrius follows Hermia and Lysander into the woods to pursue Hermia. To make matters more complicated, a young woman, Helena, trails behind her ex-boyfriend, Demetrius, because she is obsessively in love with him and jaded because he no longer loves her. Meanwhile, the spirit Puck mistakenly applies a love potion on various lovers. Once applied, this potion makes the victim love the first person they lay eyes on. Puck's mislaid potion creates mayhem between the lovers. Eventually, however, he irons out the mess he created and applies the love potion so that upon waking, Hermia will face Lysander, and Demetrius will face Helena.
In act 4, scene 1, when the lovers awaken in front of Theseus, Hippolyta, and Egeus, Theseus is shocked to see Demetrius and Lysander so peaceful, since earlier they were bitter rivals for Hermia's love. Theseus questions,

I know you two are rival enemies.
How comes this gentle concord in the world,
That hatred is so far from jealousy
To sleep by hate and fear no enmity?
(lines 129–132)

Egeus is furious to see his daughter sleeping next to Lysander. He still does not consent to their marriage. However, Demetrius, once hot-headed in his pursuit of Hermia, now claims, "my love to Hermia / Melted as the snow" (lines 152–153) and says it is Helena he truly loves. 
Moved by the genuine love of these young men and women, Theseus makes the decision to allow Hermia to marry Lysander and Demetrius to marry Helena that evening, which happens to be the night of his own wedding. In doing so, he overrides the will of Egeus and Athenian law, which states a father must consent to his daughter's marriage. Theseus states,

Egeus, I will overbear your will.
For in the temple by and by with us
These couples shall eternally be knit.
(lines 166–168)