Translation:I have seen this anime.
"I saw this anime"... It's a very language-learning kind of construction, i.e., simple but not that commonly used, so not that useful. It doesn't give any context to why you're saying it. With a time indicator, it might: "I saw this anime yesterday." That means the point of the sentence is what you dis yesterday.
The -koto- construction, by contrast, indicates that there was a time, at any point in the past near or far, where you saw the anime. I think another good translation might be: "I've seen this anime before," or "I've already seen this anime."
このアニメを見ました。 I saw this anime. (past tense)
このアニメは見たことがあります。 As for this anime, (saw it) exists. -> I have seen this anime. (present perfect)
このアニメは見たことがありました。 As for this anime, (saw it) existed. -> I had seen this anime. (past perfect)
The word こと turns the verb or adjective clause before it into a noun, similarly to の. Unlike の, it doesn't carry the sense that the activity being discussed in the subclause is at the same time as the outer clause. So it's a way of abstractly discussing situations or activities.
First, I think the second sentence should end in "あります."
Second, it seems like a different tense. Using your examples, the first is "I saw this anime" which is strictly past tense of "I watch this anime." The second is "I have seen this anime." Literally, it might be closer to a consise way of saying "I have this experience of having watched this anime."
I understand "X koto ga arimasu" as being literally "The fact/experience of X exists". The straight statement is a direct personal statement, this emphasises the fact in a slightly more polite-neutral way. Think "I saw the anime" vs "Yes, I have watched the Anime"
There are parallel phrases like "X koto ga dekimasu" - "I can X"
can anyone offer an explanation as to why this isn't ’このアニメを見ることがある？’
I feel like I had previous been taught to utilize this function in present tense - "I have the experience of watching this" or something. Maybe I am just 100% wrong, but is there is a context in which you WOULD use present tense with this structure?
You can use こと with the present tense to make a verb into a noun.
食べる (taberu) - eat
食べることが好きです。(taberu koto ga suki desu) - I like eating.
見る (miru) - watch/see
アニメを見ることが好きです。 (anime o miru koto ga suki desu) - I like watching anime.
When we talk about experiences, we need to use ta-form ことがある. We use the past form because we have the experience of having done the action in the past.
寿司を食べたことがあります。 (sushi o tabeta koto ga arimasu) - I have eaten sushi before.
このアニメを見たことがあります。 (kono anime o mita koto ga arimasu) - I have watched this anime before.
You can read more about ta-form ことがある at Learn Japanese Adventure: https://www.learn-japanese-adventure.com/japanese-grammar-experience.html
I think it's made confusing because the Japanese word アニメ can refer to both Japanese cartoons or any other cartoon. Disney movies are considered アニメ. I would say that when it comes to マンガ, Japan makes the same distinction that English speakers do, usually calling Japanese comics マンガ and American comics コミック or アメコミ (short for "American comics").
I wonder if the word "cartoon" is confusing you? By "cartoon" I mean something animated that plays on television, not like a comic strip. I can understand how these words can be strange if you don't know what they mean, but if my 70-year-old mother can tell you what both "anime" and "manga" are, then I think they're pretty commonly used in English :)
it's interesting that you're getting downvoted so much, it might have been due to how aggressive the initial comment appeared...
I've seen Japanese people on youtube refer to even cutscenes in games like metal gear as "anime" literally just meaning 'animated and not an actual camera-taken video,' no matter how realistic.