I have read in the Longman dictionary this sentence:

- I completely forgot that it's his birthday today.

I can't understand why is used the Past Simple an not the Present Perfect (I have completely forgotten that it's his birthday today ).

Both forms are correct.
The simple past shows a one-time event that is complete: I am late for his party because I forgot where I put my keys.

The perfect present shows that there is a time period during which the forgetting happened. The starting point of the period is not defined. The ending point is present point in time. (your example is good)

The past perfect is also commonly used.

I sent him a belated birthday card becuase I had forgotten that it was his birthday today.
JesusenglandI can't understand why is used the Past Simple is used ...
It's often a puzzle to non-native speakers, and yet there are many examples of it. Certain events are regarded as occurring at a definite point in time (even though that point in time is not made explicit in words); they are conceptualized in the mind of the speaker as sudden events. In these cases, the simple past is preferred (though the present perfect is not necessarily always wrong).
In this case, the speaker feels the embarrassment of forgetting as a single, sudden realization, so he says I completely forgot ....
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Hello AlpheccaStars.

Thank you very much for your help.

 CalifJim's reply was promoted to an answer.
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A correction in the sentence structure of your question:

"I can't understand why the Past Simple is used..."