How is Cassio's death supposed to inspire Othello?

Othello would be inspired by Cassio's death since he believes that Cassio was making him a cuckold by having an affair with his wife Desdemona. He would be inspired to kill Desdemona for the humiliation that he has suffered for her betrayal. Furthermore, he and Iago had made a pledge to kill the lovers, as suggested in the following extract:

... Till that a capable and wide revenge
Swallow them up. Now, by yond marble heaven,
Kneels
In the due reverence of a sacred vow
I here engage my words.

IAGO
Do not rise yet.

Kneels

Witness, you ever-burning lights above,
You elements that clip us round about,
Witness that here Iago doth give up
The execution of his wit, hands, heart,
To wrong'd Othello's service! Let him command,
And to obey shall be in me remorse,
What bloody business ever.

Iago here vows to give up everything he has to assist Othello in his revenge. This act is part of Iago's charade in manipulating Othello. He has subtly convinced the general that his erstwhile lieutenant and his wife were having an affair. Iago has now also mentioned that Cassio had been wiping his beard by using a precious handkerchief that he had given Desdemona as a wedding gift. The object has great sentimental value and significance for Othello and he believes that the only way Cassio could have it in his possession would be if Desdemona had given it to him. This, to him, is more than enough evidence that he is being cheated.

When Iago later manipulates events so that Othello actually sees the handkerchief in Cassio's possession during a conversation with Bianca, the general is even more convinced of his partner and his previous lieutenant's dishonesty. He reasons with Iago about the best way the two should be killed.

OTHELLO
Get me some poison, Iago; this night: I'll not
expostulate with her, lest her body and beauty
unprovide my mind again: this night, Iago.

IAGO
Do it not with poison, strangle her in her bed, even
the bed she hath contaminated.

OTHELLO
Good, good: the justice of it pleases: very good.

IAGO
And for Cassio, let me be his undertaker: you
shall hear more by midnight.

OTHELLO
Excellent good.

Later, when Iago wounds Cassio in the leg and he cries out in pain, Othello believes that Iago has fulfilled his part and has killed him. He is encouraged to commit his part of the promise.

OTHELLO
The voice of Cassio: Iago keeps his word.

When he again hears Cassio's cries for help, he says:

OTHELLO
'Tis he:--O brave Iago, honest and just,
That hast such noble sense of thy friend's wrong!
Thou teachest me. Minion, your dear lies dead,
And your unblest fate hies: strumpet, I come.
Forth of my heart those charms, thine eyes, are blotted;
Thy bed, lust-stain'd, shall with lust's blood be spotted.

Othello is clearly motivated by what he believes to be Cassio's just murder. He then proceeds to Desdemona's chamber where he smothers her to death.

The tragedy is that Othello's insecurity turned him int a foolish and gullible puppet in Iago's pernicious and manipulative hands. When he finally discovers the truth, he tragically takes his own life.