I was wondering, if I wanted to extend my stay for an extra 6 days, how would this happen? Some info, I'm a British citizen traveling to America with an ESTA, I'm in the state of Texas, and I've stayed currently for 25 days, I was wanting to stay for 37 now.
If you traveled with ESTA then you've been admitted under the visa waiver program (VWP). One condition of the VWP is that you cannot extend your stay or change your status. The literal answer to your question, therefore, is "no, you cannot."
You must therefore leave by the date in your I-94 record, which you can look up at https://i94.cbp.dhs.gov. That date should also be written on your admission stamp, in your passport.
VWP visitors are normally admitted for 90 days. Since you're planning to spend a little more than a month in the US, it is unclear why you think this would require extending your stay. If you indicated a shorter stay when you arrived, there's no need to notify the US government of your change in plans. If your period of admission was curtailed for some reason, then you cannot change it.