If you don't mind clobbering the array or copying it to a temporay version then you can:

use strict;
use warnings;

my @array = <DATA>;

chomp @array;
for my $index (0 .. $#array) {
  next unless $array[$index] =~ /'/ && exists $array[$index + 1];
  $array[$index] .= $array[$index + 1];
  $array[$index + 1] = '';
}

@array = grep {length} @array;

print join "\n", @array;

__DATA__
I'
m
guessing
I'
d
use
join
somehow
[download]

Prints:

I'm
guessing
I'd
use
join
somehow
[download]