My commentary in green.
Original Korean in black.
Translation in purple.

안녕하세요
Anyonghasaeyo/Hi
좋은 날 오랜만에 편지를 씁니다
It's been a while since my last letter.. writing to you on this beautiful day
‘누가 누구를 구원한다’는 말을 아주 좋아하지도, 엄청나게 믿는 편도 아니지만 과분하게도 그런 말을 가끔 듣고 살다보니 저도 당신들께 구해지지 않았나 돌아보게 되네요.
"Who saved who?" - I'm not sure I especially like or even believe in sentiments like these. But having heard words/phrases like this shared often, more than is deserved, I reflect back on whether you've all saved me too.

전보다 더 단순하게, 담백하게 혹은 담대하게 살아보려 하고 있습니다
I'm trying to live in ways more simply, more *담백하게, more boldly/courageously
감상에 빠지는 순간이 많이 줄었어요
Moments of getting lost in my emotions have decreased significantly
그래도 여전히 세상이 슬픈 건 기쁜 날이 아무리 많아도 어쩔 수 없는 것 같아요.
Still.. I think no matter how many happy days there are, it doesn't erase the ever sadness/sad things of this world.
어쩜 사실 너무 기뻐서 그런 것 같기도 하구요
Or actually... maybe it's because I'm so happy [that this is how I feel now]

*담백하게 (dambaekhagae) = something like 'simplified' or 'less complicated' specifically as it has to do with emotional baggage or overthinking/less emotionally indulgent

참 많은 말을 건넸고 많은 소리를 던져왔죠
I've shared so many words.. thrown out so many sounds right?
어떤 음률이 어떤 돌멩이가 지금 여기 계신 여러분께 닿았는진 잘 모르지만
I don't know which melodies or stones have reached you - all those of you here right now
계속 저는 그렇게 보내고 던지는 사람일 것만 같아요
I think I'll just be someone who keeps sending and throwing *those out
십삼년 전에도, 앞으로도요.
13 years ago and into the future...
들어주시는 여러분이 있다면요.
if you're still around to listen.

*those in this context referring to the "melodies (음률) / stones (돌멩이)" - which could be interpreted a lot of different ways but it's not an idiom or slang that he's referencing

함께 추억을 만들고 한 해 한 해 먹어갈 수 있다는 게 무척 기쁘고
That we can make memories together and go through season by season/one season at a time - makes me so happy
아직도 누군가에게 소리칠 수 있다는 게 좋아요
I like that I still have someone to vent (scream) to
저는 거듭 여러분을 통해 저를 봅니다
Again and again, I see myself through you all
당신도 그럴까요?
Is it the same for you?/Do you do that too?
소중한 기념일에 함께해주셔서 다시 한 번 감사합니다
Thank you for spending this precious milestone/special occasion with me.
오늘은 십삼일의 토요일 더없이 좋은 날
Today is Saturday the 13th, a day that couldn't be better
누가 그게 무슨 의미냐 해도 가슴 한 켠에 계속 우리만 아는 추억들과 기념할 날을 쌓을래요 저는
Even if others question what significance that has - I want to keep accumulating/stacking/collecting, in a corner of my heart, [with] the memories and milestones that only we share/know
모쪼록 잘 부탁드립니다
I hope to continue counting on you all
사랑합니다
Love you

HappyBirthday