I have to show the list items on edit of page and if selected some items with checkbox, how can I maintain those selected results to show only those selected records on page view.
Is it possible with JQuery-(Client Side Object Model)
If I understand your question correctly, it sounds like what you need is a new column of type Yes/No.
After you've added this column, you can choose Quick Edit to then select Yes or No (effectively the same as ticking a checkbox). Then come out of Quick Edit to see your choices saved.
If you want to see ticks and/or crosses in the list's view then you should look into SharePoint client side rendering (assuming the list view is not on a "modern" SharePoint page).