Semiconductor common emitter configuration
There's no current that must remain fixed (Why would you think that?). 

Most important feature of a bipolar transistor is that \$I_{c}\$ is controlled by \$I_{b}\$ (or \$V_{be}\$).   

\$V_{ce}\$ is regulated indirectly through \$I_{c}\$ (and thus indirectly through \$I_{b}\$ or \$V_{be}\$). Because \$V_{cc}\$ is fixed, and because there is a resistor from \$V_{cc}\$ to the collector, varying \$I_{c}\$ creates a voltage drop on the resistor which leads to voltage variation on the output side.

So in the end: Variation of \$I_{b}\$ (or \$V_{be}\$) leads to a variation of the output voltage.