So what is the solution to this dilemma? To this paradoxical, yet seemingly inevitable, nature of the relationship between humans and non-human animals, between nature and culture?
I do not have an easy answer. I am not sure if I have an answer, at all – let alone a generalisable one. I´d rather say that this is very much a question of individual ethical reasoning.
However, there is one aspect that is crucial for this kind of reasoning to be allowed and that is empathy. Empathy, i.e. the ability of putting ourselfes in the positions of others and seeing that their suffering is like our own, enables us not only to understand them better and feel compassionate toward them , but to take their interests into consideration when making decisions for our own ends, whether they be our kin, kind, a stranger or an entirely different species.
Some might ask, why we should take members of other species into consideration at all. To me the answer is quite simply, for the same reasons we take into consideration the wellbeing of every single member of the human society. We apply basic consideration to all humans, no matter of their mental or physical capacities, their character or looks. So given that we view all humans as equals in regard to those basic commonalities, I can see no reason why we should not apply the same rights to nonhuman creatures who share the same characteristics.
There are many suggestions which characteristics are necessary to imply a moral obligation on us towards someone else. But there are none I can think of that would apply to humans only and all humans, but not to any nonhuman creatures. In fact, I am rather prone to think of the one that all sentient beings have in common: The ability to suffer and a desire to live.
“In most ways, human beings are not equal; and if we seek some characteristic that all of them possess, then this characteristic must be a kind of lowest common denominator, pitched so low, that no human being lacks it. The catch is that any such characteristic that is possessed by all human beings will not be possessed only by human beings. For example, all human beings, but not only human beings are capable of feeling pain; and while only human beings are capable of solving complex mathematical problems, not all humans can do this. So it turns out that in the only sense in which we can truly say, as an assertion of fact, that all humans are equal, at least some members of other species are also “equal” – equal, that is, to some humans.” (Peter Singer) 
And if there is no objective reason to assume that human interests are in any way more important than nonhuman ones, then the only justification that remains for our treatment of nonhuman creatures is their membership of a different species – and the argument, thus, a speciesist one.
So, if we acknowledge the right to live a life free of suffering to all humans, there can be no objective reason to refuse the same to nonhuman animals.
Personally, I hope that, like with other forms of discrimination, we will cease to mistreat other sentient beings on the grounds of their species and acknowledge their place as equals. Equal, in the ability to suffer, to feel pain and pleasure, to love; and the desire to live.
 about the connection between empathy and compassion, cf. Nussbaum 2001: 327f.
 Singer (2009): 237.
Nussbaum, Martha C. (2001): Upheavals of Thought. The Intelligence of Emotions. Cambridge University Press: Cambridge, New York, Oakleigh, Madrid, Cape Town.
Singer, Peter (2009): Animal Liberation. HarperCollins Publishers: New York. First published 1975.