I will quote what contributer "pehkay" said in regards to this particular verse (reference: What exactly did Moses do wrong at Meribah?) first, and then I will add my own two cents.
From "pehkay":
The key verse is Num. 20:12: "You did not believe in Me, to sanctify
Me in the eyes of the sons of Israel" (v. 12a). This means that Moses
and Aaron had not sanctified God; they had not separated God from
themselves. Moses' speaking was wrong and his striking was also wrong.
His spirit was completely wrong, and he represented God in a wrong
way.
As God's authorial representative, Moses should have represent Him
properly. But he implicated God in his error.
God's dealing with Moses and Aaron meant that this act was committed
by Moses and Aaron only, and that God had no part in it. Perhaps,
Israel's murmuring could have been a rebellion in attitude only; their
spirit might have been different. This is why God did not judge them.
Moses should not have judged them rashly when God had not judged them.
He should not have uttered any unrestrained words according to
himself.
So when Moses makes a mistake and does not confess it, God has to step
forward to vindicate Himself.
That commentary speaks for itself, and is an excellent answer. I'll piggy-back on that and go from there (even if may be somewhat repetitive):
Upon looking at verse 8 of Numbers 20, The Lord clearly told Moses to speak to the rock, but because Moses didn't speak to the rock as he was commanded, this explains why The Lord chastised him for it, because it made it seem as though the Lord was angry with them (apparently).
I suppose the personal application for this could be: "When God tells you to do something, do it!"
Moses didn't follow directions, and for this very reason, Moses' ministry comes to a grinding halt; or, perhaps, more like it hits a bricked wall at infinite speeds! However, it's interesting to point out that even though Moses didn't follow instructions and didn't do what the Lord commanded him to do, and even though Moses misrepresented God on all accounts (hence the phrase in verse 12: "...to sanctify me in the eyes of the children of Israel,...") water still comes out, which proves that God always takes care of His children, and His mercy and longsuffering surely endures beyond mere description.
Because Moses misrepresented God, The Lord punishes them by saying, "You're not going to go into the Promised Land now. You will only be able to see it from far off."
I can relate to this, in perhaps a very minuscule way in comparison to this incredible incident, but I remember when I would do something that would warrant punishment from my mother; she would hit me where it hurt the worst, and at that time, as a teenager, I always looked forward to going out and being with my friends, who I could always see out playing from the windows where we lived. It was something that I looked forward to with utmost delight; I was always excited to go out and have fun, to "hang out" with friends, and enjoy the entire panorama of all that that was. When I would do something that warranted punishment, what would my mother do? She wouldn't allow me to go outside and be with my friends, for sometimes a week or two, depending on how serious the situation was. This was devastating to me! Not only did I want to go outside and be with my friends, and couldn't, but I would sit and watch them from the window ("from afar"), and it was the most awful, penetrating feeling.
Obviously my situation was a far cry from the incident that happened here with Moses, but in a strange way, I can relate to the feeling that this must have caused them, and I think that we all can apply something that has happened in our lives that makes this incident with Moses come 'alive' in different ways, even though this was of much greater significance.
With that said, when you synchronize these events, in Exodus 17, at Rephidim, Moses is told to strike the Rock, and of course water flows. He represented God well in that case; Moses "sanctified" God, or represented Him by following instructions.
At Meribah in Numbers 20, he was directed to "speak to" the Rock, but instead Moses strikes the Rock, which in hence misrepresents God, makes them think that perhaps (?) God is angry with them, and doesn't follow instructions, and because of this, the punishment was clear: the denial of the entry into the Promised Land.
Again: I feel that the "personal application" of this is that we shouldn't ever try to justify or crystallize away something that the Holy Spirit is telling us not to do, or vice-versa.
(On that note: Thank God for the prayers God doesn't answer...)