In American English, the article “an” is used for a vowel sound to separate the words so they don’t blend together when speaking.

Normally, “a” always precedents a consonant, while “an” precedes a vowel. But “an” also precedes vowel sounds - I.e., the sound of the letter of the beginning of a word.

An apple A banana An hour

Hour starts with a consonant, but is pronounced with a vowel sound at the beginning. Thus, it is not “a hour” and rather “an hour”.

In the case of the example from the meme, id argue that either article works:

• A “I’m…” - Typically when speaking, a person has a brief pause before they begin the quote. Since that pause would be enough to distinctly indicate two separate words, this sounds fine when being verbally spoken.
• An “I’m…” - Looks great in text and would be the correct way to list it grammatically. However when speaking this aloud, since the person would have a brief pause when saying “an” and then the quote, it probably wouldn’t sound as great to some others.

My take - I like “an ‘I’m…’” best. Both in text and verbal form. Others may disagree as far as verbally said; however, grammatically in written form this is how it should be.