Someone came to Mi Yodeya with the following problem: everyone is expected to attend an important work summit, but it's scheduled on Rosh Hashana. The person asked how to explain the significance of the day and tell the boss that attendance wouldn't be possible.
I have faced this problem several times - sometimes a holiday and sometimes Shabbat (directly, or not having time to get home). How I handle it depends in part on whether the plans can still be changed, but the broad outline is the same. It goes roughly like this:
(Name), I'd really like to be able to attend this event. (Something about why it's important.) Unfortunately, it is currently scheduled on an important religious holiday and I cannot attend. I'd like to find a way to avoid scheduling conflicts in the future; how can we work together to do that?
You want to fully participate; you value the activity. Sometimes people make excuses to get out of things they don't want to do; this is not that.
It's a scheduling conflict, not an accusation. Don't say "but you scheduled it on Rosh Hashana"; that can sound like personal criticism. This is a time for passive voice.
"Currently": if you think it can still be changed, leave that opening and ask if changes are possible.
You offer to be part of the solution. We are a minority and even if they know about our holidays they might not know about two-day days or days starting the previous day (from their perspective). At one company I maintained a calendar and included some time info when especially important (like erev Yom Kippur). Expect the burden to fall on you for a while, though they might learn in time. (After several years I changed a culture of Friday-evening gatherings at one place.)
I usually don't try to explain specific holidays unless they ask. I do explain that it's very important to observe those days and that work on those days is a violation of religious law. That's been sufficient for me so far.