In the first sentence
I am on leave after 2:30pm today.
your office coworker is saying they will be out for an extended period of time ("on leave"), beginning at 2:30pm in the afternoon.
In the second sentence, your coworker is telling you why they are out
Have a doctor's appointment for my son.
They need to take their son to the doctor's.
The listener could easily assume that both are related, and possibly your coworker's son has a serious condition since your coworker is taking "leave" (possibly "medical leave") to look after their son.
If your coworker was only leaving for the afternoon because of the doctor's appointment, they might usually say
I am out of the office today after 2:30pm.
I am leaving the office at 2:30pm.
I will be out after 2:30pm today.
It might be that their son is fairly young since
(I) Have a doctor's appointment for my son.
implies the appointment was made by the parent, instead of
My son has a doctor appointment.