It was quite near our house, but they drove very fast.

[Sarah-Betty]

I come across lost of sentences like the following that writer don't put "." when quoted sentence is finished. Could you please explain why?

Luckily my family were there, and they said, “OK, you are going straight to the hospital!” It was quite near our house, but they drove very fast.

 

[Rover_KE]

I come across lots of sentences like the following, where the writer doesn't put "." when the quoted sentence is finished.

The exclamation mark (AE exclamation point) ends the quoted sentence. An additional full stop (period) would be redundant and incorrect.

In other contexts, a question mark has the same effect.

 

[Sarah-Betty]

The exclamation mark (AE exclamation point) ends the quoted sentence. An additional full stop (period) would be redundant and incorrect. In other contexts, a question mark has the same effect.

Thank you Rover__KE for your reply.
But, I mean the following: (a full stop after quoted sentence not within quoted sentece)

Luckily my family were there, and they said, “OK, you are going straight to the hospital!”. It was quite near our house, but they drove very fast.

Not the following:

Luckily my family were there, and they said, “OK, you are going straight to the hospital!.” It was quite near our house, but they drove very fast.

 

[Matthew Wai]

While you are waiting for an answer, you may be interested to read the following: http://www.grammar-monster.com/lessons/quotation_(speech)_marks_punctuation_in_or_out.htm

 

[Rover_KE]

I have nothing to add to my answer in post #2, which emsr2d2 liked.

 

[GoesStation]

Sentence-ending punctuation within a quotation placed at the end of a sentence does double duty and also ends the sentence containing the quote. This applies even if you would have used different punctuation to end your sentence, as in your example.