It's not unusual these days to hear someone say in the same breath both that humans are distinct from the rest of the "animal kingdom," and that we must keep in mind that we are part of the continuum of life on Earth, two statements which directly contradict each other. The fact that it's not unusual, that even respected academics can still say such things without immediately being shouted down or risking their tenure, demonstrates that we are in a period of transition from religion to science. As recently as Charles Darwin's lifetime, the assertion that humans are no more or less than animals, although it was no longer particularly eyebrow-raising within biology departments, could still encounter great resistance in general in even the most progressive universities, because even then most of them were still dominated by religion. Very few universities founded more than two centuries ago were founded as other than religious institutions, and very many since have been founded as religious institutions, whose very purposes for being are based on holy texts which are thousands of years old, not on insights gained more recently which conflict with what those texts say. And, of course, universities which have been explicitly, declaredly secular cannot be expected to have been entirely immune from religious mindsets and agendas which permeate our very existence. And the notion that humans are distinct from the rest of nature is a religious notion, not shared by all religions by any means, but a central tenet of Judaism, Christianity and Islam, which teach that God put man in charge of everything and that man has a soul which is lacking in other species.
This human exceptionalism is deeply ingrained in people's minds. But biology says something completely different. It has shown us that we are not different from other animals. We're all carbon-based and we all grow and function on the basis of DNA, and humans share the majority of their DNA coding with, for example, cats. We're the same.
It's also only religiously-based thinking which causes anyone to find such plain facts insulting and/or to reject them. The religious concept of humanity being "higher" than all other life forms causes a remarkably one-sided assessment of human accomplishments when making comparisons between species. Yes, there are a long list of wonderful things which are unique on Earth to our species, the information technology by means of which you and I are now communicating being just one example. But there are also a long list of horrible things which only we humans have accomplished. No other species has produced smog or acid rain. (Not yet, anyway. We mustn't forget that all species are continually evolving, not just us.) And besides thoroughly tangible things like computers and pollution, there are assumptions made about things we don't know, such as what other species are thinking. (It's iffy enough when we claim to know something about the internal lives of other members of our own species.) Even among biologists who are atheists the assumption than no non-human species think at all, based on nothing at all but the vestiges of religious human exceptionalism, is still amazingly widespread. How do we know that dogs don't think in ways very similar to us? How do we know, for example, that they have no religious beliefs? The only rational answer is that we don't know, that such assumptions rest on primitive human mental habits and upon no firm evidence. And that we should stop making such assumptions and approach such subjects with more open minds.
Habits of thinking develop not just in individuals but also in groups, and this habit of regarding humans to be exceptional and apart from the rest of life -- again, I must emphasize, NOT shared by all humans, although it has been dominant in Western and Islamic civilizations -- this mental habit has been engrained and reinforced for thousands of years, and so perhaps it's not at all to be expected that it will vanish quickly. But we can start by recognizing where it came from, and that it has not come from science.
I don't think that there should be anything at all insulting or otherwise disappointing in seeing ourselves as animals like other animals. If adapting this attitude is a negative thing for you, perhaps you don't know non-human species as well as you could and don't love them nearly as much as you could.