After you click 
Run and connect to CrossBrowserTesting.com, TestComplete will launch the environments you have specified, copy your test project to them and command TestExecute installed in those environments to run the specified test.
You can monitor the test progress in the CrossBrowserTesting manager:
Click the image to enlarge it.
The manager shows the state of each environment and the state of each test in each environment.
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See Also
Running Tests in CrossBrowserTesting Environments - Tutorial
Integration With CrossBrowserTesting.com