Two sets are "the same size" is better said as "two sets have the same cardinality" when there exists a bijection from one to the other.
But essentially, the "size" of a set is the "cardinality of the set". Better yet, speaking of size, especially when speaking of the size of infinite sets, countable or otherwise, is a question of the cardinality of the sets.
Cardinal arithmetic acts pretty much as we would expect when examining the cardinality/size of finite sets. But cardinal numbers and cardinal arithmetic, specifically when equating or comparing the cardinality of infinite sets, is not necessarily what we would "expect", intuitively, until you have a firm grasp of what cardinality represents.
For example, the interval/subset of the real numbers given by the interval $(0, 1)$ has the same cardinality ("size") as the set of real numbers! There does, in fact, exist a bijection from the interval $(0, 1)$ to the entire set of real numbers. Here too, it seems very unintuitive to say so, which is why a thorough understanding of what cardinality is (and what it is not) is crucial.
As an aside: you might be interested in this post: A set is infinite if and only if it is equivalent to a proper subset of itself. Here, "equivalent to" means "has the same cardinality as".