Big brains equals high intelligence and Bigger brains equals higher intelligence. You must have seen that I crossed the equal word in the statement because that statement isn't always true. No doubt humans have big brains compared to a lot of animals and we are quite intelligent but we do not need to increase the size of our brain to increase our intelligence.
Studies have shown that brain volume only accounts for 2% of variation in performance during cognitive tasks and so it is not a useful tool to compare how smart a person is to another but then our brain size as a specie has increased averagely and it till fits our head. The size of the body of an animal or specie doesn't determine its brain size although the bigger the size of an animal the bigger its brain but when comparing body size with relation to brain size, smaller animals tend to have big brain in relation to their body size and if we are to judge based on that, it means that larger animals tend to have smaller brain in relation to their body size.
Humans for instance have a brain to body ratio of 1:47 while the elephant has a brain to body ratio of 1:1165. Now, if you as an animal is going to have a big brain, then you are going to be needing a big head and brains require a whole lot of fuel for energy and burn a lot of energy with the brain using between 2 to 10% of an animal's daily energy but this is even quite more in humans. Our brains make up 2% off our body mass but accounts for 20% of our total energy used daily.
For humans, we cannot explicitly say how our brains have evolved since brains cannot be fossilized since they degrade over time but we have learned through studying the fossils of previously existing cousins with endocast in their skull that our brains have increased in size over the years and this increased brain size also comes with more complex neural connection. Which gave rise to humans being able to do a lot of things from hunting to socializing.
For our brains, its intelligence in relation to its size has been explained using the Expensive Tissue Hypothesis which suggests that the brain takes energy from other bodily systems such as the digestive system in other for it to grow. This was seconded by a published study in 1995 that found that our brain was too big for our body size and our guts were smaller for our body size. Talking about gut size, herbivores have more digestive organ space compared to carnivores and this is the same for primates as the ones that eat leaves tend to have bigger guts than the ones that don't.
We have read a lot of hypothesis supporting why we have big brains and a lot of these hypothesis have been linked to our behavior and social lives but in the last 3000 years, we have seen human brains shrink by 4 ping pong balls of volume. Scientists are trying to understand why this is happening with so many guess flying round. Some propose it could be as a result of climate warming, others propose it could be a product of complex social living. In all, the once big brain we tend to have in relation to our body mass is gradually shrinking.
https://pmc.ncbi.nlm.nih.gov/articles/PMC4305162/
https://www.open.edu/openlearn/science-maths-technology
https://www.frontiersin.org/journals/ecology-and-evolution/articles/10.3389/fevo.2022.963568/full
https://pmc.ncbi.nlm.nih.gov/articles/PMC3046985/
https://nyaspubs.onlinelibrary.wiley.com/doi/10.1111/j.1749-6632.2011.05988.x
https://www.sciencedirect.com/topics/veterinary-science-and
https://pmc.ncbi.nlm.nih.gov/articles/PMC5498304/
https://www.journals.uchicago.edu/doi/abs/10.1086/204350
https://www.frontiersin.org/journals/ecology-and-evolution/articles/10.3389/fevo.2021.742639/full
https://karger.com/bbe/article/98/2/93/835670/Climate-Change-Influences-Brain-Size-in-Humans
https://www.bbc.com/future/article/20220503-why-human-brains-were-bigger-3000-years-ago