The problem can be divided into smaller tasks and distributed across multiple computers. These computers communicate over a network and work together to solve the larger problem.
IBM - What is Distributed Computing?
Not necessarily. Computers need to communicate and coordinate their work, which takes time. Some problems are also harder to divide than others, so performance does not increase perfectly with every additional computer.
Yes, as long as fewer than one-third of them are faulty. This is the Byzantine Generals Problem: honest computers must agree even when some crash or send false messages. To handle f bad computers, you need at least 3f + 1 in total, so 1,000 computers can handle up to 333 liars. The tradeoff is that agreeing takes many rounds of messages, which makes it slower.
Source - Byzantine Generals Problem
Much of the computation happens on remote servers in data centers, rather than directly on your phone or computer. Your device sends requests over the Internet, and these servers process the data and send the results back.
Cloudflare - What is the Cloud?
We could build much more detailed climate simulations. According to the source, older climate models see the atmosphere at about 100 km resolution, while exascale supercomputers can simulate it down to about 1 km. This lets researchers predict regional weather extremes far more accurately. Getting there takes an enormous number of computers working together as one reliable machine, because a failure partway through would ruin a long simulation.