I don't think your status should play any role here. You've published a few papers yourself, and have a reputation of your own based on your work. It is off of that reputation that you've been asked to peer review, the same as most other peer reviewers. So, I wouldn't even mention it unless asked, in which case be truthful but don't show doubt. Editors are also usually pretty aware that you are junior, and often they select a mixture of reviewers, eg. one junior and one senior.
If the situation makes you feel a little insecure as to whether you're able to deliver as good a review as someone more senior, then you can also ask others for input, but this mustn't reveal the paper you are reviewing to outside parties. For instance you could ask your advisor "I'm reviewing a paper for journal X, the author wrote that ...., do you think it is appropriate for me to say that they should do this and that?" (it can be delicate to try not to reveal too much here, but it depends a little on how much trust there is between your advisor and you). It's also perfectly fine to query the editor in charge: when I reviewed my first paper I was unsure whether it was OK to bring up a specific issue and how to bring it up without sounding like I'm overly negative, so I emailed the editor and he gave me some good advice. Good thing is in that case you don't have to try and keep the paper under review a mystery. You can also ask the editor to have a look over your final draft of the review before handing it in, maybe mentioning that it was one of the first reviews you've done and if he thinks it could be improved before handing in the final thing.
In your PhD application this should go on your CV, under the Service heading. This can be task oriented, eg a sub point like this :
Peer review: Journal X, Journal Y, JZ
Or it could be chronological:
2017 - Reviewer for Journal X
2016 - YZ Committee member
The point of these kinds of entries is twofold: they work as esteem indicators (they show that other people respect you enough to ask you to do certain things that require trust/ability), and they show that you're willing to pull your weight in the academic machine (something commonly known as "collecting cookie points" or "brownie points").