"I have seen this anime."
Well... sort of. Depends on the context.
Regular past tense (-た or -ました) is generally used when there's an implication of a specific point in time. Ex: if you're talking with someone about what you did last week and you said 「映画を見た」, it would be understood that you were talking about watching a movie last week.
On the other hand, sometimes you need to express that you did something in the past without implying a certain time period. That's what -たことがあります does. It's basically saying "at some point in my life, could be days or years ago, I did this thing." Or, more succinctly, "I have experienced doing this thing."
You'll also often hear this as a question (-たことがありますか？), which translates to "Have you ever...?" A very useful phrase to phrase to remember!