the 's' before the regular expression is referred to as a pattern-matching modifier, and denotes that the expression is a substition/replacement expression.
the '#' is just acting as a delimiter. common delimiters are "/" or "#" or "{}" or "[]" typically
s#/+#/#
=
s/\/+/\//
When you use # instead of / as the delimiter, you make a pattern matching expression that would normally use the default delimiter, like the example, simpler, since you don't have to escape it
The patter above is simple:
match the "/" character 1 or more times, as many as possible without encounter some other token and replace that match with a single "/"
s#^(/project\d+/)#/projects/#
or
s#^/project\d+/#/projects/#
or
s/^\/project\d+\//\/projects\//
matches a string where, at the beginning is starts with "/project" followed by 1 or more numbers and ending with "/" such as "/project1234/" and changes it to "/projects/"
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