The major key problem that drives and links the scenes in the whole play is jealousy. The two major characters, showing the opposite of each other, the hero Othello and the villain Iago are both controlled by jealousy, changing from the "Lord" position into a "monster" driving each other's position down, exactly as mentioned in the quote and therefore it should be considered as the key quote of the play. .
To understand this quote better we should first consider the meaning of the word "jealousy" and "green-eyed monster".
"Jealousy is a state of fear, suspicion, revenge or envy caused by a real or imagined threat or challenge to one's possessive instincts. It may be provoked by rivalry, in sexual love, by competition or by desires for the qualities or possessions of another" according to The New Lexicon Webster's Encyclopedic Dictionary.
"Green is a colour of many unripe foods which cause stomach pains." According to the website http://phrases.shu.ac.uk. Shakespear often uses green colour to describe jealousy because its taste is very similar to jealousy. Here because many cats have green eyes and often play with mice, their food before eating them, similar action as when one is jealousy of another often plots traps acting as a "monster".
The power of jealousy can be seen from the very start of the play up to the very end, the downfall of all the major characters. The victims of jealousy appear from the very start of the play. One of the most clear character is Roderigo, who loves Desdemona very much and is able to do anything to gain her love. His jealousy to the love between Desdemona and Othello almost drive him mad. His mind is absolutely eaten by jealousy. This leads him to believe Iago and to follow his plans, whereas at the end he fails attacking Cassio and killed by his most trustful friend - Iago. .
For the same case, jealousy of love can be seen in Othello. The love which drives him mad, which creates "fear, suspicion, revenge, envy caused by an imagined threat" provoked by his "rivalry, the desires for the possessions of another" - Iago, who noticed and uses his weakness, jealousy as the target.
