This is more a question about terminology than English.
In trading, stock exchanges have set trading hours. For example the New York Stock Exchange trades from 09:30-16:00.
When financial reports say "shares closed at..." it is referring to their value at closing time.
So in your example:
Chandler, Arizona-based Insys had no immediate comment. Its shares closed up 7.6 percent at $7.36 on the Nasdaq
I understand that to mean that when trading closed, shares for Chandler were "up" (that is higher than when they began trading) by 7.6%.
To answer your last question: you would not write that shares went up to a percentage because that doesn't mean anything mathematically. A percentage is not an amount of money, so if I said something had "gone up to 7.6%" it is meaningless and you still don't know what it is worth. Saying "gone up by 7.6%" is a reference to its original monetary value.
I would say that grammatically it doesn't matter if you say "up by 7.6%" or omit the word "by" - but as we are talking about trading terminology the normal rules of English may not apply. I do not work on the stock market, but if your original quotation was lifted from a financial publication I would say that is normal to simply say "up 7.6%"