Learn Turkish: Repliğimi unuttum ve stresten ağlayacak gibi oldum. - I forgot my line and felt like I was going to cry from stress.

The literal translation is closer to "I became like I will cry." In Turkish, the structure -(y)AcAk gibi olmak is the standard way to express being on the verge of doing something or experiencing an imminent sensation. While English uses "felt like," Turkish uses olmak (to become) to describe entering that physical or emotional state, so you wouldn't use hissetmek (to feel) here.

Why does the sentence use the ablative case in stresten instead of a phrase like stres yüzünden (because of stress)?

The ablative case suffix -DAn (here as -ten due to consonant harmony) is frequently used to show the direct cause of an emotion, action, or physical reaction. It translates perfectly to "from" or "out of" in English (like "crying from stress"). You could technically say stres yüzünden, but stresten is more concise and sounds more natural for expressing the direct trigger of an emotional state.

What are the suffixes doing at the end of repliğimi?

The root is replik (a line in a play or script). First, we add the first-person possessive -im (my line), which causes the final k to soften into a ğ (repliğim). Then, the specific direct object suffix (accusative case) -i is added because "my line" is the specific, defined thing that was forgotten (unuttum). This gives us repliğimi.