Learn Turkish: Ayrımcılık ile ilgili zihniyetini garipsemiyordum. - I didn't use to find your mindset about discrimination strange.

The base word is the adjective garip (strange). By adding the attitude/perception suffix, we get garipsemek, which means "to find strange" or "to consider strange." Note that while the suffix is often categorized broadly as -(I)msEmEk (like in benimsemek or küçümsemek), it sometimes takes the shorter form -sEmEk, as it does here in garipsemek.

Why is there an -i- in the middle of garipsemiyordum instead of an -e-?

This is an example of vowel narrowing caused by the -yor suffix. The stem plus the negative suffix is garipseme-. Because Turkish doesn't allow a wide vowel (a or e) directly before -yor, the -e in the negative suffix narrows to an -i, making it garipsemiyor- rather than garipsemeyor-.

The translation says "your mindset." Could zihniyetini also mean "his/her mindset"?

Yes, absolutely. Out of context, this word is completely ambiguous. It can be parsed as zihniyet-in-i (mindset + your + accusative) or zihniyet-i-n-i (mindset + his/her + buffer letter 'n' + accusative). We only know it means "your" here because of the English translation provided.

Why is ile ilgili used here, and could I use instead?