It might help to think in terms of memory addresses instead of variable names.
my @a = (); # Set aside memory address 123 for a list.
my $a_ref = [@a]; # Square brackets set aside memory address 456.
# @a COPIES the stuff from address 123 to 456.
push(@$a_ref,"hello"); # Push a string into address 456.
print $a[0]; # Print address 123.
The string went into a different memory location.
Instead, point the $a_ref variable to the memory location of list @a. push affects memory location 123. Since @a also refers to memory location 123, its value also changes.
my $a_ref = \@a; # Point $a_ref to address 123.
push(@$a_ref,"hello"); # Push a string into address 123.
print $a[0]; # Print address 123.