Sneaking something with

()(ぬす) means to act while someone is not watching. Depending on the context, it can mean “behind someone’s back” or “slip past”.

Literal Meaning

A literal translation is to “steal the eyes”.

Usage Notes

The person acts while someone else is not watching, usually to avoid detection or punishment. Depending on the act, the phrase can sound mischievous or underhanded.

In 〜の目を盗んで, the noun before の names the person whose notice the subject avoids.

Dictionary Definition

(ひと)()つからないように、こっそりする。

Do it secretly, so that people don't find out.

Goo Dictionary

Further Reading

Examples

Sneaking sleeping pills in with 目を盗む

Learn the meaning and usage of the idiom 目を盗む in Japanese, with examples from manga.
名探偵コナン » Volume 9 » Page 154

As Ran sleeps, she’s carried to a sink filled with water and almost drowned. After, Conan considers the coffee Ran drank earlier must have been drugged, or else Ran would have fought back. Since Mrs. Yone, who prepared the coffee with Ran’s help, was with Kogoro at the time of the attack, Korogo narrows the list of suspects to four who were out of sight while the coffee was being prepared.

小五郎(こごろ):
「という(こと)は、あなた(かた)(なか)(だれ)かが(よね)さんと(らん)()(ぬす)んで睡眠薬(すいみんやく)()れたとも(かんが)えられますな?」
“Considering that, isn't it possible one of you slipped past Mrs. Yone and Ran and put sleeping pills in?”