おっしゃる
ossharu
Their words go on the high shelf, yours go on the floor
The picture Words coming out of the other person's mouth, caught in both hands and set on a shelf one level above where your own words sit. おっしゃる puts their words on the high shelf; 申す sets your own words down on the floor.
Where the picture shows up in おっしゃる
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copying what's on the shelf
部長がおっしゃったとおりにしました。
buchō ga osshatta tōri ni shimashita
I did it exactly as you said.
Taking the manager's sentence down off the shelf and tracing it by hand, line for line.
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holding out an empty slot
お名前は何とおっしゃいますか。
o-namae wa nan to osshaimasu ka
May I ask your name?
One slot on the shelf cleared and held toward the other person's mouth, waiting for a name to be placed in it.
One level above 名前は何ですか — standard on the phone and at a counter.
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checking if anything was left up there
先生は何かおっしゃっていましたか。
sensei wa nanika osshatte imashita ka
Did the doctor say anything?
Looking up to see whether the words that left the doctor's mouth are still sitting on the shelf.
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pointing at the shelf to agree
おっしゃる通りです。
ossharu tōri desu
It is exactly as you say.
An open palm indicating the shelf where their words rest, with a nod.
おっしゃる通りです is the most common way to agree in a meeting.
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asking for it to be set up again
もう一度おっしゃっていただけますか。
mō ichido osshatte itadakemasu ka
Could you say that once more?
The words on the shelf too faint to read, so both hands go out asking for them to be placed up there again.
All of it in one picture
おっしゃる doesn't alter a hair of what 言う means; it only raises the shelf those words rest on. Take a sentence down off that shelf and copy it and you get "I did as you said"; hold out an empty slot on the shelf and you are asking someone's name. Point up at the shelf and nod and it becomes agreement; when what's up there is too faint to hear, you hold out both hands and ask for it to be set up again.