I am curious if this thing is possible to do on page layout. If i select some value from picklist/checkbox the text field should show up on page layout.
This is currently not possible using native Salesforce without a Visualforce Page. But there is an Idea with a whooping 54000+ points raised long back but in consideration now. Latest update says that the Product Team wants to address this Idea but the timeline is more than a few releases out. https://success.salesforce.com/ideaView?id=08730000000BroxAAC
Please vote for this idea.
If you are looking for Appexchange Solutions, May I suggest you to go with Skuid which is a powerful UI related App. Relevant Link : http://help.skuidify.com/m/supercharge-your-ui/l/173514
The only way I can think of doing this on a standard layout is by using a picklist/checkbox field along with a workflow rule to change the recordType of the record and have different layouts for each recordType, one with and one without the text field. Otherwise, you could embed a custom visualforce page and use the rendered attribute to evaluate the picklist/checkbox field to determine whether or not that text field should display. Available on both input and output textfields.
rendered Boolean A Boolean value that specifies whether the component is rendered on the page. If not specified, this value defaults to true. 10.0 global