I just thought it was weird that a massive group of people would decide that somehow, crossing this ice sheet would lead to a better world on the other side.
My history teacher suggested that they were following animal herds, but then that raises a further question, why on earth would animals cross an ice sheet? It's not like the land bridges were just a couple of miles wide and that you could see grass and trees on the other side.
I might be asking a really stupid question that has an obvious answer, but this really has been bugging me for days, and I thought you guys might be able to help me out.
This may surprise you, but Beringia was much different than you're imagining! Despite its popular portrayal as a frozen hellscape, Beringia was actually a relatively ice-free place to be during the period known as the Last Glacial Maximum (LGM). Here's a helpful diagram^2 showing the relative extent of the glaciation, although it omits some small sheets in Eastern Siberia. As you can see, it's pretty limited aside from the glaciers cutting Beringia off from the rest of North America. I'll discuss the LGM here because while humans may have inhabited the area prior to this, they were almost certainly in Beringia by the time it ended.
Most of the region was probably covered in a biome that closely resembles what we know today as tundra. We even have some idea what the weather would have been like during this period, with winters being dry and bitterly cold (approx. -10°F through -40°F ^1) and summers wet and cool (with the warmest months averaging approx. 50°F to 60°F). Regionally, the warmest parts of Beringia would have been those in the west, in modern Siberia. Farther east, the weather would have been a bit colder, with what's now Anchorage and the Yukon positively frigid.
So, let's get back to your original question: why would people cross an ice sheet? As you guessed, they wouldn't! Indeed, they almost certainly couldn't have even had they wanted. Instead, paleoindians would have followed the caribou and other large game across the landscape, using their skins for clothing to survive the cold (yet liveable!) environment.
^1 Kuzmin, Y. V. (2008). Siberia at the Last Glacial Maximum: environment and archaeology. Journal of Archaeological Research, 16(2), 163-221.
^2 Meyer, V. D., Hefter, J., Lohmann, G., Max, L., Tiedemann, R., & Mollenhauer, G. (2017). Summer temperature evolution on the Kamchatka Peninsula, Russian Far East, during the past 20 000 years. Climate of the Past, 13(4), 359.
The Bering Land Bridge was not an ice sheet, but a wide grasslands steppe (about 1000 kms/600 miles wide at its zenith) teeming with animal life, and populated by people for thousands of years. Most of Beringia, as it's called, is now underwater, but archaeologists, geologists, paleontologists, paleontologists etc. have learned a lot about that lost world from the islands and coastal remnants of Beringia.
The ice sheets you're thinking about were on the continent of North America, and - depending on the date of entry - would have presented a challenge for entering North America from Beringia. There are a lot of arguments over the route the first people took into the Americas, and exactly when they entered. But no theory involves people going over a massive ice sheet. The old theory, which was nearly taught as fact when I was a kid in elementary school (despite being shakier in academic circles), was that around 10,000 years ago people entered via an ice-free passage between two ice-sheets through the interior of Western North America.
However, earlier dates for settlement of the Americas have emerged, and an alternate entrance theory: that people entered the Americas by following the Pacific coastline in boats is very popular today. Jury is still out on a lot of questions, as you'll find if you go down this fascinating rabbit hole.
I recommend checking out this discussion between /u/RioAbajo and /u/Reedstilt on this subreddit over the peopling of the Americas.
And for Beringia itself, it's a bit frustrating to see that the book Human Ecology of Beringia by John F. Hoffecker and Scott A. Elias is quite expensive, but it's good.
Simon Fraser University has a site on the Beringian migration: A Journey to the New Land, which has subsites for elementary school students all the way up to post-secondary resources. It presents the history around these theories and archaeological sites in the North in a very accessible way.
(ETA: If you're ever up to the Yukon, the Beringia Interpretive Centre in Whitehorse is a must-visit. It's a small museum in an out-of-the-way corner of the world, but a wonderful glimpse into that lost world.)
Maybe ask AskAnthropology?
This one is a little out of the realm of History.
