This has been discussed as a poorly done jump scene. Since the birdcage is only seen quickly in the background in an earlier scene and it's never a focus, either on film, or in dialog, when we jump to it after the fairly explicit love scene between Leticia and Hank, most viewers are left only scratching their head.
What I understand about that scene, is the reference to a caged thing. In the beginning of the story, we see the death row inmate, Leticia's husband, Lawrence. He is the caged thing. During his death walk, he is led to the chair by Hank, his men, and Hank's son Sonny, in his first death walk. His son disrespects the solemnity of the occasion, by throwing up, mid-walk. When it happens, Lawrence is concerned for Sonny, and asks about his well-being. The thing is, however, in a death walk, it is imperative to keep everything low key and solemn, otherwise the convict about to be killed may begin to fight back and struggle against the death. The officers want everything under control at all times. In the case of Leticia's husband, his death shows no love, no personality or emotions.
The convict is 'released' from all his Earthly concerns. He no longer has to deal with racism, hate, crimes, victims, loved ones... he has been freed from his cage.
Later on, without realizing who she is, as related to the convict he just walked to the chair, Hank starts a relationship with Leticia. Every other act of sex, for him, was just like the death row... never looked the prostitute in the face, no kisses, no love, no emotions of any kind, just sex.
When Hank and Leticia have sex in her living room, after her son's death, she is very emotional herself, exacerbated by the whiskey.
At first, Hank 'takes' her the same way he took the prostitute, from the rear, no looking her in the face, no kissing, no real emotion. But Leticia lost everything, her husband, her son, soon her house (she has to know how far behind she is with rent). She needs the closeness, to held. So she gets up, and turns around, sitting back down on him, but now facing Hank, where she can hold him, and in turn, be held, and kissed. And loved, at least for a minute.
This is a first for Hank. This turns sex into love. Even though she's black, he realizes that he is feeling more than just sex. He gets her emotions and starts the journey away from a loner racist who lost his "artist" son, who has an even worse racist for a father. He starts to feel the release of his own emotions and prejudices.
Then, in Forster's jump cut, we see the caged bird. We also see both hands, his and hers, reach in, and take the bird. We assume they freed the bird from its cage, just as they've been freed from their own problems.
Note, this is my own interpretation, which may be different than yours. Doesn't make yours any more or less accurate, because the only one who knows, is the director, and he never discussed it. I read the screenplay (it's great, and actually VERY readable, compared to most) and the scene is there, but no reference to the cage at all.