Website Down issue – Memory Leakage

My site recently got this problem “Error establishing a database connection” When i contact with my hosting service they reply me that

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The above logs describe that your website coding/scripts will open MYSQL Connections and will remain open and these threads will cause your website slow response and in the end your website showing MYSQL Database connection error.

How can i fix this error i am not a good programmer any one please help and explain Thanks in advance.