Translation:My older sister is good at swimming.
It captures an entire sentence, ending with a verb or adjective (verbs behave very much like adjectives in Japanese), and turns that situation into a noun. (Called nominalization.)
A subtlety is that it does this in a way which generally refers to that situation occurring at the same time as the outer sentence it's used in. It will especially always be used with sensory verbs like 見る or 聞く to explain what you see or hear, etc.
There is a similar word こと which you'll also run into quite a lot. It also turns the preceding clause into a noun, but more abstractly, not suggesting that it's going on at the same moment as the overall thing you're saying. For example it will get used along with 約束する to turn the thing being promised into a noun.
You are correct. The only thing I can think of is thay some females might be offended to be called big as it might imply fat. But most younger siblings would call the elder sibling their "big sister" or "big brother". I think the real reason for the difference here is that the Japanese course is very new and has a bunch of inconsistencies in it.
True. The difference is that およぐの is a modified verb meaning "the act(ivity) of swimming" (in whatever way), while すいえい is a noun that's mostly used in reference to the sport of "swimming". The latter can also be turned into verb by adding する though and sounds a bit more formal/serious.