I may trying to convert the array that I get from mysql to variable. Then array passes though my function node, the payload is unchanged ans still has the array as the payload.
You are assigning the converted value directly to the message object as msg.FullName or msg.name.
If you want to see it in the payload, you have to assign it there.
Normally you could just assign the result set to msg.payload, like
msg.payload = msg.payload[0] to get an object like {FullName: 'testfullname', TalkTitle: '...'}...
Those strings shouldn't be buffers in the first place, which data types are you using for these fields? That shouldn't happen with varchar or char, they should be normal strings.
If you want to keep it that way, you need to convert each one and assign them manually.