What's This?

Setting a so called "reward recipient" means that you define a beneficiary who will get the BURST reward in case should you (= your miner) find a block.
In case you are mining with a pool, you enter the pool's address. In case you are solo mining you enter your own address.

Reward recipient:

You have to enter here the passphrase of your wallet (more precisely: the wallet you are mining with), because setting a reward recipient is effectively a transaction, so you have to sign it.

Your secret passphrase:

After hitting "Submit" this will happen:

If everything went well...

If everything goes well you will get some output in JSON format in your browser. It will be a block of text that will have a bunch of transaction details like this...

{"signatureHash":"77765d5f8929f67e752c7yaddayaddad9c1bc7190a0a79e6ae876717b770","unsignedTransactionBytes": "14101eb97605a00552c52de00b69b239a13b7b11b6a92f5bead44cfe83fc415b41c8d07a2e07831d03be5aff840881c4000000000 yadda yadda yadda {"senderPublicKey":"52c52de00b69b239a1etceteracfe83fc415b41c8d07a2e07831d","signature": "4355bd909049928615fe6541594fd46blahblahblah37eaede052e0cd208330c93b41d29929a74658d5f89d337ab39c66de8e1874 fc02533157c3989","feeNQT":"100000000","type":20,"fullHash":"c23ddc18786bd89e2a

Yeah - it looks like an error, but it's not. :-) Now for the most important part:

You will now have to wait 4 Blocks (16-20 minutes on average) until your change takes effect!

Optional: To confirm that the transaction went through, you can go to your wallet and check your latest transactions. You should see a "Reward Recipient Assignment" transaction with the account of the Pool's Burst address.

If something went wrong

The JSON message will be much shorter and contain a terse error message. In this case check if your passphrase is correct or if you haven't made some error in the reward recipient address. Or both.

For another nice explanation what's happening here see this.