Assuming you are using CBC within either Lighthouse Studio (or SSI Web), open your project that includes your CBC exercise. Go to File + Data Management...
Click, AddJob... , and then click Add again. Select the file type "CBC (*.csv , .cho/chs, .att)".
On the File Format option, click "Sincle Format CSV". This will put all the attribute level information shown to respondents in each CBC question along with the respondents' answers within the same .CSV file.
Click OK. Then, Click Export Jobs. A .CSV file is written to a file folder within your project.
This .CSV file has your data arranged as 27 rows per respondent (assuming you have 9 choice sets with 3 options per choice set).
To get respondent-based demographic characteristics also in this file, you will need to export your respondent characteristics separately using the Data Management area. Then, you'll need to do some extra data processing to put respondent characteristics in the CBC .CSV data file that has 27 rows per respondent. I don't know how Stata is supposed to accept those data, so you'll need to refer to their documentation.