Hi @veeraraghavanraghava, welcome to our forum and congratulations on your new Xperia X!
Do you know how the contacts were stored in your Xperia L? For example, if you have them synchronized to a Google account or some other Online account? If they are stored on a Online account, you should be able to add this account to your new phone and they will be synced to the phone.
If the display is not working on your previous device, you will not be able to access any data on the device as this require the device to be unlocked.
If you used a SD card in your Xperia L and saved the contacts there, try inserting it in your Xperia X and go to the Contacts application > 3 dots in the top right corner > Import/export > Import from SD card of internal storage (.vcf file).
I hope that helps!
