In Act II, Cassio cries, “I have lost the immortal part of myself, and what remains is bestial.” How does this sentence apply to Othello?

Because Othello similarly loses the immortal part of himself, leaving something far more uncivilised, savage, animalistic and  bestial.

What do we mean by the immortal part of himself? Well, it could mean lots of things. The part of you which doesn't die. Is that the soul? The religious awareness - the bit that goes to heaven? Or is it just simply a person's goodness?

In any of these cases, it's obvious that, as Othello submits to Iago's suggestions, and believes that his good, noble, angelic wife has cuckolded him, the veneer of his Christianity vanishes into a bleaker more fundamentally bestial worldview. "Why did I marry?" he asks, and - by the end of the play, he is calling for "blood", and raving about "goats and monkeys".