1. What does it mean?
As slang, 오글거리다 describes the squirming embarrassment you feel while watching or recalling something excessively sentimental or theatrical. It can be directed at your own old words as well as somebody else's.
Also written or said as 오글, 오글거려, ogeulgeorida
2. The nuance
It can be affectionate teasing, but it can also hurt when someone is being sincere. The reaction does not prove that a confession, poem, or emotional scene is bad; it describes the speaker's discomfort.
3. Examples
“옛날 자기소개 글을 다시 읽으니 좀 오글거려.”
Rereading my old introduction makes me cringe a little.
“대사는 오글거리는데 두 배우가 진심으로 연기해서 좋았어.”
The dialogue was cheesy, but I liked how sincerely the two actors played it.
Sources
KoreanClass101: Korean language lessons
References support the meaning and usage. Example sentences are written for NoCapWiki.