How can we know or measure pain in animals?
Pain is subjective

Pain is a subjective experience; you cannot even tell with certainty how much pain your fellow human is experiencing, which is why we ask people; they then can tell us. Pain relief (both physical and emotional) is a significant part of medicine, yet we still have "pain scales" for self-reported pain, one of the more common ones being the Wong-Baker Faces Pain Rating Scale:



Now, note well that even this scale must be interpreted for the patients. For instance, a patient might look like a 6, but be reporting a 10, in which case, a nurse must try to ascertain their actual level of pain. 

My point is evaluating pain is contentious even in humans who can express themselves.

Definition of pain

The International Association for the Study of Pain (IASP; www.iasp-pain.org) defines pain in humans as 


  an unpleasant sensory and emotional experience associated with actual or potential tissue damage, or described in terms of such damage.


Pain usually involves a noxious stimulus that activates nociceptors in the body that carry signals to the CNS where these signals are processed (and generate responses) including the “unpleasant sensory and emotional experience.”

The following are examples of common types of noxious stimuli for different tissues:


  
  Skin: thermal (hot or cold), mechanical (cutting, pinching, crushing), and chemical (inflammatory and other mediators released from or synthesized by damaged skin, and exogenous chemical stimuli such as formalin, carrageenan, bee venom, capsaicin)
  Joints: mechanical (rotation/torque beyond the joint’s normal range of motion) and chemical (inflammatory and other mediators released into or injected into the joint capsule)
  Muscle: mechanical (blunt force, stretching, crushing, overuse) and chemical (inflammatory and other mediators released from or injected into muscle)
  Viscera: mechanical (distension, traction on the mesentery) and chemical (inflammatory and other mediators released from inflamed or ischemic organs, inhaled irritants). 
  


Animals need us to interpret their pain into language. For political and emotional reasons, humans are ill-disposed to do so (how could we then justify eating, exploiting, experimenting on animals, etc.). Even into the late '80's, veterinarians were taught that animals didn't perceive pain per se.

How we know animals understand pain

To determine whether animals can experience actual pain (not simply nociception), it is necessary to show that they


  can discriminate painful from nonpainful states; make decisions based on this discrimination in a way that cannot arise from evolved nonconscious nociceptive responses; demonstrate motivations to avoid pain; and display affective states of fear or anxiety if threatened with noxious stimuli. In addition, animals experiencing pain might be expected to exhibit spontaneous behavioral changes including sustained signals of distress and impairments in normal behaviors such as sleep.


Animals show all these behaviors. Additionally, there is ample evidence that emotional pain in animals is real. The mere threat of foot shock (electric) induces signs of stress in rats and mice that can be alleviated with anxiolytics (drugs that reduce anxiety).

Even fish show responses to a painful event: guarding behaviors, unresponsiveness to external stimuli and increased respiration, all of which improve with morphine. This is called analogous evidence: evidence that they are, mechanistically at least, directly analogous to pain responses in more complex animals. 

Humans are animals

The nervous system of mammals is fairly identical to our own (we are animals). A chimpanzee who is afraid of a pin after being stuck with one 20 times in a row clearly feels physical pain and fear. That a chimp can mourn the death of a parent by not eating and not moving for days clearly indicates that they can experience emotional pain. 

In summary:

Pain is subjective. I cannot really tell exactly what you feel if your arm is ripped off (your bird-wing example) or (don't read if queasy type)


   if you are given an electric shock, your throat is slit and, while still bleeding, you are dipped in boiling water to remove your body hair in a whole body hair plucker (which is how most poultry - usually males, not hens, btw - meet their makers). 


By definition, only you can tell us.

But common sense and experience tells us it would be painful. And, yes, animals feel pain pretty much exactly like we do. Because we are exceedingly "more alike" than we are "slightly different".

Can animals feel pain?
Pain in Research Animals: General Principles and Considerations
Which Responses Indicate Pain and Which Nonhuman Vertebrates Display Them?&lt;- read this if you really want to know
Evolution of pain