So, I am reading Steve Awodey "Category Theory", and what he claims is that if category has finite products and equalizers then it has terminal object. I am not so sure why is this true. I mean, he says "a category", which can be large. So, a terminal object is a product of all objects in a category, so possibly infinite products also have to exist. Why is this true? Thanks!
The terminal object is not the product of all objects, it is the product of no objects. Already having finite products implies having a terminal object, by taking the product of the empty set of objects.
Why is the terminal object the empty product? See this question.
"Finite products" includes products of families of objects of any finite size... including zero.
The terminal object is the product of the empty family; i.e. a family of zero objects.