According to wiki, mammoths were usually around the size of the Asian elephant, so they weren't really as unusual in size compared to dinosaurs.
In the past, some mammals achieved almost dinosaurian proportions. The
Songhua River Mammoth, for example, reached a height of 17 feet and a weight of almost 20 tons, compared to 10 feet and 6 tons for a modern day African Elephant. The
Paraceratherium were even larger, being over 18 feet tall and perhaps up to 30 tons, putting it in the same range as some sauropod dinosaurs.
I believe there are two main reasons why dinosaurs were so much larger then modern day animals; one, the food they ate and the ecosystems they lived in encouraged it. The plants and vegetation of the Mesozoic era were much different; angiosperm plants and broad leafed trees didn't exist during most of this era, and the dominant plants were ferns and conifers. These are hard to digest, so herbivorous animals needed to eat more of them and have large stomachs and digestive tracks to digest all that food. This necessitated that these animals become larger. Carnivorous animals needed to become larger as well, so that they could hunt the plant eaters.
The Mesozoic was a much warmer time then our current age, which allowed a much more robust ecosystem. There was much more food to support a much higher number of large animals. Which brings us to my second point;
In the last several million years, we have been living in an ice age cycle. The polar ice caps have been growing and retreating in a continuous cycle every few hundred thousand years or so. Even during a interglacial period, in which we are currently in, the Earth's temperature is still much cooler then it was say, twenty million years ago.
The cooler temperatures only allow a fraction of the large animals that flourished in the Mesozoic ecosystem.
Also keep in mind that humans have been putting extra stress on the ecosystem pretty much the entire time we have existed. Before humans, animals the size of elephants such as Mammoths and wholly rhinos were pretty common, and lions (and similar big cats) were spread across all of Europe, Asia, Africa and the Americas.
As far as the oxygen level argument goes, I believe that oxygen level has a much larger effect on arthropods then vertebrate animals such as dinosaurs and mammals. Consider the oceans; many of our modern whale species are much larger then any dinosaurs, or even any of the animals that lived in the Mesozoic oceans. I believe that this suggests that oxygen rates aren't a decisive issue here.
If you brought a Brachiosaurus through time to the present, for example, I don't think it would suffocate due to the oxygen level. It might be somewhat more sluggish, but I think its biggest problem would be finding enough food to support itself in our colder ecosystem.