The last part of your post might misled people to think that our is a must for removing the syntax error. So just a little bit more detail...

Remove the package qualifier, but keep local, it results syntax error:

use strict;
use warnings;

package Foo;

local $bar;  

sub init {
    $bar = 42;  
}

1;
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Global symbol "$bar" requires explicit package name at Foo.pm line 6.
Global symbol "$bar" requires explicit package name at Foo.pm line 9.
Global symbol "$bar" requires explicit package name at Foo.pm line 10.
Foo.pm had compilation errors.
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But use either my or our removes the error:

use strict;
use warnings;

package Foo;

my $bar;  

sub init {
    $bar = 42;  
    print $bar;
}

1;
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But the use of our or my does make difference in other sense. The following script works when $bar is defined with our:

use strict;
use warnings;

use Foo;

Foo::init();
print $Foo::bar;
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But does not work if $bar is defined with my:

Name "Foo::bar" used only once: possible typo at math1.pl line 7.
Use of uninitialized value in print at math1.pl line 7.
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my or our? depends on what you want to do - in what scope you want your variables to be seen.