When you refresh the screen, you see "Transfer in Progress" as shown with three of my domains I transferred:

If everything checks out, then you are done. The next step will be to wait for the transfer to be complete.

Troubleshooting

If however you get something like I did with one of my domains, you will see something like this:

Possible and most likely scenario is that you did not copy the code correctly. Repeat step 3 by re-entering your domain name and go to your email that NameCheap sent you showing you what your authorization code is. Recopy and re-paste the code into the box labeled Transfer Code as you did in step 8 and click on Add Domain button.

You will see a new line appear with your domain name saying "Ready to Transfer". Click on the Green Button as you did in Step 9.

You will have to re-do the payment process as shown here:

Follow through and you see the following screen where you see the second attempt to transfer your domain has been registered and WA is attempting to validate the verification code.

You will also be re-sent the email from WA as you did in Step 12. Go through the same steps such as agreeing to the terms as you did in Step 13.

Once the agreement form has been submitted, your updated domain transfer screen should look like the screen below and the updated domain listed should read "Domain awaiting transfer initation"

If this does not solve the problem, you may need to contact NameCheap or submit a support ticket.