This happened to me several times in a row today. I pressed F5 to Debug my Windows Azure project in Visual Studio 2013 with the emulators, and the project would build, but no browser launched. There was no build error either, it would just complete the build and look like it was ready to launch a browser, but then just stop.
Finally I saw a message in the bottom left status bar of Visual Studio that said "Compute emulator shutdown". Then I moused over the Azure icon in my taskbar, and it said "Storage Emulator is started", but "Compute Emulator is shutdown".
I was helped by this post that mentioned where the emulator log files were. The key log file I needed was
C:\Users[User]\AppData\Local\dftmp\EmulatorRuntimeLogs\ErrorRuntime.log
That log file had an error in it about failing to write a file. This reminded me that I had forgotten to start Visual Studio as Administrator. Its a simple step for any Azure project but something that I forget to do sometimes since the majority of projects that I work on in Visual Studio are not Azure projects. And I didn't see this specific failure case mentioned in Stack Overflow, so thought I would post it.
I closed Visual Studio, right-clicked on the Visual Studio 2013 launch icon, chose "Run as Administrator", and then browser launched fine when Debugging