=:= tests if two containers (variables or items of arrays or hashes) are aliased, ie if one changes, does the other change as well?
my $x;
my @a = 1, 2, 3;
# $x =:= @a[0] is false
$x := @a[0];
# now $x == 1, and $x =:= @a[0] is true
$x = 4;
# now @a is 4, 2, 3
As for the others: === tests if two references point to the same object, and eqv tests if two things are structurally equivalent. So [1, 2, 3] === [1, 2, 3] will be false (not the same array), but [1, 2, 3] eqv [1, 2, 3] will be true (same structure).
leg compares strings like Perl 5’s cmp, while Perl 6’s cmp is smarter and will compare numbers like <=> and strings like leg.
13 leg 4 # -1, because 1 is smaller than 4, and leg converts to string
13 cmp 4 # +1, because both are numbers, so use numeric comparison.
Finally ~~ is the “smart match”, it answers the question “does $x match $y“. If $y is a type, it’s type check. If $y is a regex, it’s regex match – and so on.