1. to look at something carefully, examining, ensuring everything is correct and as it should be
to look at/examine something with a critical eye
If you observe or review something with a 'critical eye' you are in a negative frame of mine and looking for the negatives in the work or situation, rather than positives. This may allow you to point out mistakes or play 'devil's advocate'. You might ask someone to take a look at your work 'with a critical eye' in order to get honest feedback, rather than fishing for praise. This is an idiomatic phrase used primarily in educational and professional environments.