I'd say her name, they'd get upset = I say her name, they get upset??
SCRIPT
I don't even see Dana. I just, I see, I see the blood, and my mom picking me up, and putting me on the bed, and whispering that it's not my fault. But after that, every time I'd say her name, they'd get upset. I wasn't even allowed to say it anymore.

QUESTION
every time I'd say her name, they'd get upset
- I wonder that "every time I say her name, they get upset"
- Does it mean same as the sentence right above?
If not so, is there any slight or big difference?
Please explain me about this. Thanks in advance ^^!

Question

I'd say her name, they'd get upset = I say her name, they get upset??
SCRIPT
I don't even see Dana. I just, I see, I see the blood, and my mom picking me up, and putting me on the bed, and whispering that it's not my fault. But after that, every time I'd say her name, they'd get upset. I wasn't even allowed to say it anymore.

QUESTION
 every time I'd say her name, they'd get upset
-> I wonder that "every time I say her name, they get upset" 
-> Does it mean same as the sentence right above? 
If not so, is there any slight or big difference?
Please explain me about this. Thanks in advance ^^!

Answer

I'd say her name, they'd get upset = repeated actions over a period of time in the past
every time I say her name, they get upset = repeated action in a period of time around the present

So, no, they don't mean exactly the same.

Answer

every time I'd say her name, they'd get upset.= every time (whenever) I took her name, they became upset. and yes, it can also mean the action and reaction going on till now too. 
After that comment, it is very natural to ask, even now?
ans: yes, even now. (this will clarify if the action-reaction happened in past only or still happening nowadays )
Just another way of saying.

Answer

I say her name.... is a form that retells the story of what happened in the past as though it is happening now.

I'd say her name... Keeps the story in the past as it is retold.

They both work because a script should read like how people talk.  If it were not a script, I would say I'd tell a story is more correct.  But characters do not always say what is most correct.  You choose.

Answer

OMG! Hahahaha What a foolish guy I was. Thank you for commenting this, Peachey!

Answer

Just a comment correction: "(Could you) please explain this to me."  Your sentence asks the reader to explain you, not the situation.  Also, "explain" needs an object, so no preposition.