This is one of the reasons I practice atheopaganism. It's psychologically helpful for me to be tuned to the natural, annual cycle. It gives a rhythm to life that's separate from the more artificial, civil-labor thing. It also inspires gratitude for some of the little complexities of nature I wouldn't otherwise notice.
For me it's my annual fishing trip. It's never exactly at the same time, but we always do it. Sometimes it's only 2 of my closest friends from high school, but us 3 never miss it
I like tracking constellations and Mars as they move through the sky throughout the year. I've gotten really good at knowing where exactly to look for specific constellations at different times of year. I know it isn't impressive, but I really like the fact that I can always find the stars I adore.
I also love going out and looking at any celestial events I hear about. I woke up at 3 AM once and stood outside in 20F weather to watch an eclipse. So worth it.
My aunt gave me my great grandfather's stargazing binoculars last year and they're so nice. My great grandfather worked for NASA and was one of the most intelligent people I've known. He died before I was old enough to really appreciate his brilliance and accomplishments, but I strive to learn about him. Owning and using his binoculars makes me feel closer to him in a way. He and my aunt are the people that got me super into space and stuff. My biggest dream is to go to space, but I won't ever go unless cheap and safe space tourism that allows people with chronic illness somehow becomes a thing in my lifetime.
This is the kind of thing I think about to get the useful bits out of the "touch grass" meme.
Unfortunately I'm not very good at it and time isn't real until the stores put out the commercialized holiday crap: It's only really summer when the 4th of July kitsch is put out on the shelves. It sucks and I want to be better about it.
It helps to be more in tune with your surroundings outside. Many of us become disconnected from the land around us because of the pace of modern life: working 40+ hours, driving everywhere, and generally not having energy to go outside in our spare time. But even taking a short walk around the block each day will let you get in tune with your local climate. You'll learn what the temperature is like when the first flowers start to bloom, when foods are in season, when the frost starts to come.
Unfortunately, this may also mean you start to notice how out of whack the cycles have become thanks to climate change.
While I'm sure it's not this guys intention, a "nature" calendar is also why the calendar was invented to begin with, specifically for growing crops. Another incarnation of the nature calendar is the Farmer's Almanac.
For me it's the first good bit of right crunchy cronch of snow underboot. We only had it for maybe a week so far this winter, but it was still nice to take it in while it lasted.
Then there's the crish crush crunch of autumn leaves which are just the right crunchiness to crack like a crispy chip.