As with most things in life, showerheads can and do wear out. It happens that they can develop cracks or defects, and this results in a Showerhead Leak.
The simple solution is to simply replace the showerhead. Before you replace it, take the showerhead off, then put it back on, perhaps it was not installed properly to begin with.
However if the leak has developed over time then it is likely the showerhead has simply worn out. This is one plumbing that most of us plumbing novices can handle.