.=method is a mutating call of the method on an object. The call $x.=method does the same as the more verbose assignment $x = $x.method.
In the following example, the $o container initially holds an object of the C class, but after $o.=m(), the value will be replaced with an instance of the D class.
class D { }Â class C { Â Â Â method m() { Â Â Â Â Â Â Â return D.new; Â Â Â } } my $o = C.new; say $o.WHAT;Â # (C)Â $o.=m(); say $o.WHAT;Â # (D)