There are two approaches to this. Most here are interpreting this using assuming the most scientifically sound version, that animals can only communicate up to their level of sapience, but most tv/literature instead shows human-level speech from animals instead. This is a very important distinction. Humans effectively provide moral weight to level of sapience. Killing a human or anything with human level sapience is unforgivable, killing a monkey or high order sapient mammal is the worst of animal quality, killing a chicken is acceptable, but only for a good reason, and killing an insect is done just because. The more sapient a creature is the more it is presumed to deserve human-level protection of law and culture.
Animals capable of Human level communication
This is scientifically improbable, to say the least, but this is how animal speech is usually implemented in most literature I see. If you went with this approach it would completely revolutionize how we treat all animals. I hope you like being a vegetarian, because killing and eating something you were just debating philosophy with is not going to happen. There would be a huge fight, but animals all over the world would receive far closer to human level protections. Pets would still exist, probably even occur more often, and some mostly-human use of animals, like riding of horses, may still exist; but only as a job the horse chooses to be employed at for pay in oats.
Animals still limited to personal sapience
This is far more realistic, though still hard to implement. Interpreting would be far easier side to do, but you wouldn't get nuanced sentences from most creatures. Look at your average dog, an experienced dog trainer can read quite a bit in body language, they are a very communicative species, However, they don't use verbs and adjectives and complex syntax. Your translator may say "defending home, uncertain if your threat" or "afraid and submitting" or something along those lines, explaining the meaning of their body language, but there will not be any real nuance beyond that, because that is as far as their minds are capable of parsing and thus expressing.
Talking back to the animal is far harder. Dog communication is mostly body language, a machine can't really communicate body language. Other creatures may throw in pheromones, or sound above or below what is audible to humans, trying to build a machine capable of doing any of those is quite difficult. It would almost be more plausible to assume some chip was implanted in the animal's brain that short circuited the process and jumped right to sending the concepts to their mind (still doesn't make sense, but the issues with this approach are easier to hand wave away by saying "we just learned allot more about the brain then we know now".
In terms of how it would change us, not too much for most animals. It may make pet owners better pet owners, many owners do not know how to interpret their own animals communications. Some may see an animal groveling in fear and think it's a cute thing they are doing to entertain their owner, without knowing the real meaning. Way too many allow small dogs to be rather aggressive and dominate because they think it's cute as well. A better understanding of what their pets are actually 'saying' could make them better owners, and perhaps lead to more pet owners. It would also make any industry that requires communication with animals easier, like breeding or riding horses.
Morally I don't think our views will be challenged by most creatures. However, there are a certain group of highly sapient animals that we constantly are setting harder and harder challenges to meet the definition of sapience because we want to believe humans are completely unique. These animals include many apes, octopus, dolphins, and any number of other 'smart' animals you have heard of.
these creatures still wouldn't be capable of full human-level communication. It's been shown they can't comprehend some concepts, such as the symbolic use of images (a scale model can represent a larger object etc). However, they will prove to have pretty extensive communication ability, we have already taught many apes to communicate rather well with sign language or symbol-boards.
This will of course prove a boon in researching these creatures, but the more interesting question is how humans will respond to the questions about their sapience and thus how deserving they are of protection. A believe a higher degree of animal rights activist would show up and argue for protective status of theses creatures, great ape personhood is already a movement now, though one that is mostly ignored.
However, there will still be plenty who will not agree to with this. They will point out the obvious limits to what these creatures can do to communicate. They will argue that the device you are using are putting 'words in their mouth' and is not an accurate translation, giving the appearance of more sapience then exists. They will argue that even if these creatures can communicate at something above toddler-level language they still lack creativity or innovation etc. In all honesty the majority of humans will just not notice one way or another, there are no apes local to them so they see it as academic and don't really care.
I imagine this would lead to some interesting philosophical debates and attempts to better define what gives one the right to legal protection. However, how extensive these debates actually became I'm not certain, as I aid the majority of the 1st world nations are far removed from most of these creatures, and in all honesty only 1st world nations will be able to afford expensive communication devices and philosophical debates on personhood, when your a subsistance farmer all you care about is keeping animals away from your crops you grow, and possible supplementing your food with some bush meat.