Step 1 – Click Me at the top of your LinkedIn homepage, then select Settings & Privacy from the dropdown.
Step 2 – Click to the Privacy tab and scroll down to the Getting a copy of your data (under the How LinkedIn uses your data section):
Step 3 – Tick the connections box and hit Request archive.
In about 10 minutes or so, you’ll get an email with all your connections’ email addresses in a handy spreadsheet.
This method should be used responsibly to prevent you from losing your LinkedIn account.