Birds are strongly linked Thomasin Yeobright in the novel.
In her movements, in her gaze, she reminded the beholder of a feathered creature who lived around [Mrs. Yeobright's] home. All similes and allegories concerning her began and ended with birds. There was as much variety in her motions as in their flight. (3.6.23)
Looking at Thomasin in terms of this symbol can help us get a better grasp on her character, which is otherwise a bit hard to pin down. We don't see Thomasin a lot in the book, even though she's one of the major characters. So how can birds help us understand her?
Well, birds have both negative and positive ideas associated with them. Let's start off with the negative, so we can end on a happier note. Birds are associated with ideas of cages and entrapment, and also vulnerability (birds are often delicate and fragile). We see evidence of how Thomasin is trapped and made vulnerable by her ill-advised marriage to Damon:
"I want some money, you know, aunt [...] and he doesn't give me any. I don't like to ask him, and yet, perhaps, he doesn't give it to me because, he doesn't know. Ought I to mention it to him, aunt?" (3.6.29)
Themes and imagery of entrapment appear through birds in another notable scene as well, though this one involves Thomasin's aunt and not Thomasin herself, directly at least.
The only living thing that entered now was a sparrow, and seeing no movements to cause alarm, he hopped boldly around the room. [...] This roused the lonely sitter, who got up, released the bird, and went to the door. She was expecting Thomasin [...]. (3.7.1)
What's interesting is that this sparrow is trapped and lost in Mrs. Yeobright's home, just as Mrs. Yeobright herself is, in a way – she is sort of isolated through her own stubbornness in this scene. But the bird also helps to draw Mrs. Yeobright out of her own depression and gets her to take some action. In a way, this bird has a liberating effect on her, since it "frees" her from her morose thoughts, temporarily at least.
This brings us to the more positive aspects of birds: freedom and movement.
Feathered species sojourned here in hiding which would have created wonder if found elsewhere. [...] A traveller who should walk and observe any of these visitants as Venn observed them now could feel himself to be in direct communication with regions unknown to man. [...] But the bird, like many other philosophers, seemed as he looked at the reddleman to think that a present moment of comfortable reality was worth a decade of memories. (1.10.1-3)
We see here how birds have a sort of positive, even otherworldly, energy associated with them. Birds prompt Diggory to think about looking for happiness in the present and to ponder things beyond his immediate surroundings, much as the sparrow did for Mrs. Yeobright.
The birds also subtly represent Thomasin here as well, since Diggory's past is largely about Thomasin and his present and future happiness hinge upon her. In a way, Thomasin, like these birds, might have a sort of emotional freedom and inner calmness that comes from her practicality and her ability to live in the present and in harmony with nature, like the birds.
Overall, birds are largely feminine in the novel and the opposing things they represent, entrapment and freedom, really capture the difficult status of women in this novel and in the Victorian era.