You don't have to fly to Thailand at all if you don't want to.
The entire application process is online. You apply through the BOI on their website
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The application process can take 1-3 months, depending on how complicated your situation is.
Once you are approved, you will receive an endorsement letter that is valid for 30 or 60 days, I forgot which one.
They will then ask you, do you want to apply for your visa through your Thai embassy online, or do you want to go to Bangkok to pickup the visa and have it applied into your passport?
If you want to get it from the embassy, you will apply online with your endorsement letter, and pay whatever your embassy charges, likely more than the 50,000 baht stated as the price. Your visa will then be a printed pdf document that you have to carry for 10 years along with your passport.
If you want to go to Bangkok to get it, you will need to make an appointment with the BOI to go into their office at One Bangkok and they will take you over to the immigration office where you will pay exactly 50,000 baht and receive the visa in your passport, so you won't have anything extra to carry around.