The staff represents that what he was doing was empowered by Hashem because it had Hashem’s name on it
Since Torah is life, when Moshe, a Tzaddik, taught it Torah, it caused the force of life in it to come out, enabling it to respond to Moshe and let water out.
Rock, release the water!
Hashem said הסלע twice, showing that it can be with any rock.
No! It can only be Miriam's well.
Do it on this Rock! This one!
The staff is for hitting the rock which would cause the water to come out because it had Hashem's name on it, which would awaken the potential in the rock, causing it to listen to his instructions.
קח את המטה
Then, one should instruct the rock to release the water
ודברתם
Hashem said הסלע twice, showing that it must be that specific rock and no other.