"The birds finish their food."
Translation:הציפורים מסיימות את האוכל שלהן.
Because the noun ציפור is actually feminine. Hence the feminine use of the verb, and feminine possessive at the end of the sentence.
It is an irregular noun in that it doesn't have the typical feminine ending (e.g., ה-) in the singular. It also is irregular in the plural, in that it doesn't have the usual feminine ending of ות-.
Many irregular nouns, like שמש/sun, are irregular in only form and regular in the other, in this example, the plural for suns שמשות has the usual feminine ending.
There are some clues/rules for irregular gender nouns, e.g., most body parts that come in pairs are feminine, as well as countries, but some irregulars you just have to learn/know.