Learn Turkish: Biz civcivleri kümese götürüyorduk. - We were taking the chicks to the henhouse.

The -i is the accusative case ending. It shows that civcivler (chicks) is the specific direct object of the verb. Because it has this ending, it translates to 'the chicks' rather than just general 'chicks'.

What does the -e do at the end of kümese?

It is the dative case ending, which indicates direction towards a place. It changes kümes (henhouse) into kümese (to the henhouse).

How is the verb götürüyorduk built to mean 'were taking'?

This is the past continuous tense. It combines the verb root götür (take), the present continuous marker -üyor, the past tense marker -du, and the personal ending -k (we). Together, they show an action we were in the middle of doing in the past.

Could I use getiriyorduk here instead of götürüyorduk?

Usually not. Getirmek means to bring something here, towards the speaker. Götürmek means to take something away to another place, which fits perfectly when you are moving chicks from your current location to the henhouse.