It doesn't really "do" anything except assist how Perl interprets the expression.

# say would think that its arguments are the whole statement in bracke
+ts
say +(split ':')[0] while (<DATA>);

#..so we use the + to make it clear that the contents of the brackets 
+are not function arguments
# but part of an expression
say +(split ':')[0] while (<DATA>);
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It's there for syntax disambiguation. The + starts an expression, which gives the compiler a clue to not treat the parens that follow as if they were part of the syntax of a function call parameter list for say.

The examples in perlfunc demonstrate this syntax quite neatly:

Any function (...) may be used either with or without parentheses around its arguments. (...) If you use parentheses, the simple but occasionally surprising rule is this: It looks like a function, therefore it is a function, and precedence doesn't matter. Otherwise it's a list operator or unary operator, and precedence does matter. Whitespace between the function and left parenthesis doesn't count, so sometimes you need to be careful:

    print 1+2+4; # Prints 7.
    print(1+2) + 4; # Prints 3.
    print (1+2)+4; # Also prints 3!
    print +(1+2)+4; # Prints 7.
    print ((1+2)+4); # Prints 7.
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It would have been useful if you'd posted the original code. Here it is:

use Modern::Perl;

say +(split ':')[0] while (<DATA>);
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say (and the same goes for print which has fuller documentation) can take a FILEHANDLE as its first argument.

say FILEHANDLE LIST
print FILEHANDLE LIST
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Edit: Sorry, that last paragraph was poorly worded. Hopefully, this next one is better. Also see the example code in the Update at the end.

If the FILEHANDLE is omitted, the '+' indicates that the arguments following the '+' (once expanded) are the LIST to be output. If the FILEHANDLE is included, the '+' is not required.

More complete details can be found in print and perlop - Symbolic Unary Operators.

Regarding Modern::Perl, that will effectively do the use feature 'say'; for you; if you comment that out, you will need to do it yourself. I think that's what you were getting at; although, I did get a little lost in the multiple negatives and conditions: "the code works if ... but it doesn't unless ...".

Update: I felt my response might have been misconstrued as being contrary to the previous responses. This is certainly not my intention; in fact, there weren't any responses when I wrote mine. The links to documention that I provided do give more detailed information; however, the following code examples may clarify what I was saying.

$ perl -Mstrict -Mwarnings -E 'my @x = (1,2,3); say (@x)[1]'
say (...) interpreted as function at -e line 1.
syntax error at -e line 1, near ")["
Execution of -e aborted due to compilation errors.
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$ perl -Mstrict -Mwarnings -E 'my @x = (1,2,3); say STDOUT (@x)[1]'
2
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$ perl -Mstrict -Mwarnings -E 'my @x = (1,2,3); say +(@x)[1]'
2
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Thank you, Anonymous Monk, for asking this.
I was actually looking for this syntax, having seen this before and I was getting weary of all the parantheses I'd need without it.

Cheers, Sören