Of course the world would change XD
You've taken a fundamental rule "Animals Can't talk" and transformed it. The world would fundamentally change from far simpler changes like making 10 animals talk.
What do they say? There is a great line between making words and being able to use them in conversation and understanding more complex things. If we have 'increased' the intelligence of a dog in the particular ability for him to talk... well that just means he'll be able to make it to about a 3-4 year old level of understanding by the time he's about to die of old age and the dog will most likely always be unable to perceive or understand certain concepts, even some that humans consider basic.
With this sort of 'intelligence', animals all over would become more protected, and we would probably all go vegetarian as farmers felt guilty when the cow could scream out "No!". But we wouldn't grant most animals a very high place in society or anything. A few animals, those that are already considered the most intelligent, might get granted 'personhood' by some governments. That is, they would be given full rights. I doubt full rights of humans would be given to all animals though. The most intelligent of these animals could even learn to work at complicated jobs. Your calls might get directed to a pig instead of a man in India when your computer is freezing up. Of course, if they can work, they can also get a job instead of you, or collect unemployment...
But of course, this brings into question what you meant by 'animals'... Insects are considered animals by scientific definition, so does that mean the ants on the ground can say hi to me or scream when i accidentally squash one of their legs? the Flies in the air?
Mankind will soon grow use to hearing the screams from our day to day activities, and I suspect mankind's entire system would break down, with millions dying. Those who do survive would probably become use to the noise of the animals and stop caring. This might even, ultimately, make a humanity where the strongest and the majority are even less moral towards the screams of other humans as they have grown use to the screams of the animals. There would, of course, be some minority group which went all monk and survived trying never to hurt a fly, but they would be as large as the monks are today... that is tiny and barely hanging on to existence.
Oh wait... Maybe I took this thread too seriously. XD