This important allusion that the monster makes about himself comes in Chapter Fifteen, as he describes to Victor Frankenstein precisely how he came to learn speech and to read and learn about his own situation. As he reads various texts, he identifies himself with various characters in them. Key to this process is his reading of Paradise Lost by John Milton. As a result of this he identifies himself as Adam at first, but then feels that identifying himself as Satan is more appropriate for his position. Note what he says:
Many times I considered Satan as the fitter emblem of my condition. For often, like him, when I viewed the bliss of my protectors, the bitter gall of envy rose within me.
The comparison is clear. The creature, by his physical appearance, is naturally excluded and isolated from the feelings of love and companionship that the humans he watches are able to enjoy. Like Satan, therefore, he has been created with the ability to feel those emotions and to desire them, but his appearance and character makes enjoying and experiencing those emotions himself impossible. Thus he is envious as he witnessess the felicity of the De Lacey family.