Animals are a lot more like us than people realise. Just like us, animals also like stability and comfort in their lives. They can get stressed or worrisome. They can express happiness which is apparent to those who understand it when they see it.
Fear, I believe, is one of the most primal emotions any animal can have to aid in its survival. Those who say animals lack any sense of fear or pain are lying, because they must know it is necessary for survival. Do they feel it on the same level as we do? Probably not because they may lack the complex reasoning we have and the ability to see and confront what they fear on a deeper level. But it's for certain that most animals feel it. Imagine an animal faced with a source of danger and pain, and they know it will bring danger and pain, such as fire or a predator. They avoid it like their life depends on it, the adrenaline starts pumping, they pull away and cry out when they are hurt by it, they may become panicked and frantic - just like us when we are faced with immediate danger in those cases where all our human logic and reasoning go flying out the window due to panic. Fear, panic, anxiety are all feelings, no matter how shallow an animal's fear or panic may seem from our perspective.
Seeking pleasure, I believe, is also a very primal instinct, and happiness a very basic primitive emotion. Whether or not animals are able to understand want and happiness the way we understand it, it is clear that animals seek what brings them pleasure and avoid what brings them pain. That too aids in survival, because the things that make them happy (food, shelter, a fertile mate) are also things that will help them survive and pass on their genes. If you give an animal a reward for an action, it will continue to perform that action because it wants the reward again. It is seeking something that brings it pleasure. Pleasure is a feeling, even if it is very shallow from our perspective.
Deeper and more complex emotions are a bit harder to define, because the way an animal expresses them may not be anywhere near recognisable to us because they express them in such a different way and may also have emotions that we do not have any reference for (while also lacking emotions we have, causing a lack of common ground which makes interpretation difficult). There is a lot of research, for example, into chimpanzee language. They are physiologically, genetically and psychologically the most similar animals to humans. Yet we struggle to interpret their complexities. At a glance, chimpanzees appear savage and mindless. With careful observation, it eventually becomes clear they actually have a very structured way of interacting with one another and complex expressions. Complex expressions could indicate complex feelings, but it has taken a long time for people to realise by just how much and we're still not finished figuring them out yet.
If we invest so much energy into understanding the possible emotions and expressions of our closest genetic relative, having changed our opinion so much with that research, it becomes clear that differences in expression and behaviour between species are difficult to interpret to such a point where people can misread and underestimate what animals are capable of. What looks like mindless animalistic behaviour to us might be discovered later on to be a very complex and purposeful behaviour, reflecting deeper feelings held by the animal.
I doubt though that we will ever fully understand. Just like complex behaviours can indicate complex feelings, they can also be entirely instinctive with no emotion behind it aside from the simple pleasure (reward) of carrying out an instinctive behaviour.
