You have to remember that this is poetry, and poetry often breaks a lot of the rules of the English language.
Envy means that you're jealous of someone. This poem is about jealousy. The writer was born of a chimney sweep and an oyster-wife. This means that the father was very poor and regarded as a very lowly person, but the mother was a pearl (the valuable stuff sometimes found in oysters) so the child (the writer) is torn between two realities. As the child of a well-regarded mother, the writer feels that he or she should have the finer things in life, such as an education, expensive clothes, and rich food. However, because of the father, the child (the writer) was denied these things and is envious (or jealous) of everyone else who does.
Because the writer can't read, he or she wants all the books to be burned. That way nobody else could read either.
The writer goes hungry because he or she can't afford enough food due to poverty, and is jealous of everyone else who has enough to eat. Rather than go hungry while everyone else is fed, the writer would rather that there was a famine and everyone died of hunger. If everyone else was dead, it wouldn't matter how underfed and skinny the writer is, he or she would still be eating better than everyone else.
The sentence "must thou sit and I stand" refers to servants serving nobility. The nobility sit in luxury, while the servants are forced to stand so as to be able to dote on them more quickly.
The final line "come down, with a vengeance!" is the writer's wish for revenge. The writer wants everyone who seems to be better off in terms of wealth, opportunity, etc. to be brought down, or abased, insomuch that they are lower even than the writer himself/herself.
I hope this helps!