What makes Shakespeare's Much Ado about Nothing entertaining?

The play is entertaining because it is a comedy about love with an element of drama. 

Most of the entertainment value from the play comes from the squabbles of the lovers Benedick and Beatrice.  Both of them claim to be not interested in a serious relationship, but they enjoy bickering and they have been in love before.  When Beatrice’s cousin Hero plans to wed Claudio, it seems to spark their interest. 

Hero and Claudio supply the play’s drama.  They are supposed to marry, but on the wedding day Claudio condemns Hero as untrue.  The play’s villain, Don John, has convinced him that he saw Hero cheating on him with another man.  It was actually not Hero, but Margaret.  Beatrice protests this slander, but the men, including Hero’s father Leonato, seem to accept what Claudio says at face value. 

The play also has comic relief in the form of Dogberry, the hapless constable.  He is hilarious, because he is completely inept.  He mixes up his words and does not do a very good job investigating the case of Don Pedro and his minions. 

Watchman

If we know him to be a thief, shall we not lay
hands on him?

DOGBERRY

Truly, by your office, you may; but I think they
that touch pitch will be defiled: the most peaceable
way for you, if you do take a thief, is to let him
show himself what he is and steal out of your company. (Act 3, Scene 3)

Shakespeare is playing on words here, of course, with "thief" and "steal."  The audience would have found it funny, even if Dogberry does not mean to be so.  In the end, everything works out and Don Pedro is punished for trying to ruin Hero’s reputation.

Most entertaining plays will have a satisfying ending.   In the end, Hero and Claudio are able to marry after all, so that crisis is averted.  Beatrice and Benedick are also in love.  It is a tidy little wrap-up.