The Arabic language contains “time modifiers”–words like “before” and “after”. In Arabic, these are called Duruwf Zamaan, and grammatically, they act like the possessive case.
While there are tons of these in Arabic, there are two you run into pretty often in the Qur’an and ahadith: ba’da (after) and qabla (before).
Grammatically, the modifier comes before the thing it modifies–the same as in English. And since the modifier acts like the possessor in the possessive case, the modified receives kasra.
So for example, “the duck left after the cow” would be “kharajat al-battatu ba’da al-baqaratiِ.” Notice the time modifier is the possessor in a possessive case–hence the word immediately after it (the possessed) takes kasra.
As another example, “he left before today” would be “dhahaba qabla al-yawmi.” Again, the time modifier is the possesser, and “today” is the possessed–so it takes kasra.
Also, note that the words that ARE duruwf zaman all have fatha on the last letter–so it’s ba’da, qabla, and so on.
That’s it; it’s quite simple really. If you feel you don’t understand, go back and read the possessive case post, then read this one again. Post a comment if you have any questions/comments/clarifications/etc.