As someone who recently (this past year) went from being a renter to a homeowner, and had previously been a home owner for several years, it's not all rational... but here are some of the reasons.
I don't have to worry about the rent going up every year / six-months. If I plan to own the house for multiple years, a large chunk of my money goes to equity: I'm banking away value for later that you will get back when you sell the house. (Assuming the market doesn't go pear-shape: a big assumption.) In contrast, money you spend on rent is a pure cost, and irrevocably lost.
There are several "rent vs buy" calculators that can visualize (based on interest rates and rent vs mortgage costs) the crossover point where it's cheaper than it is to rent for X years, and what it does for your net worth. For me, it appears that about 2-3 years of owning will be a break-even point, and less if my rent would have increased.
In my case, the mortgage on a house (3 bedroom) + yard + garage, with a nice kitchen, was about $100 more per month than a cramped, 2 bedroom apartment with a miniscule (and poorly designed) kitchen, no space for the kids to run around, and not enough space to even unpack our stuff.
I OWN IT. Until you've owned one, it's possible you might not fully comprehend this double-edged sword.
I can paint the walls, break the walls, rip up the floor, put in a new mailbox, take out the trees I don't like. There's nearly nothing that keeps me from renting out a bedroom to a friend, or letting the in-laws stay for a week. If I want to mount a swamp cooler outside my window, I can, or turn my garage into an archery range.
There are downsides, too. As an owner, you risk that the market will fall out from under you. It's harder to move, as you have this limbo of not being able to buy a house until you've sold the previous one. (I haven't figured out how to make this not suck. Same goes from transitioning from a lease to ownership.) I have to mow the lawn, make sure the garden is watered. The utilities are a bit more expensive.
Also, as a homeowner, you tend to accumulate More Crap, since you have more space to fill.
All of this is worth it, though, as I do not worry about whether it's OK to paint my kids' rooms, or tear out a ceiling fan that I hate, or stain the deck a different color, or completely redo the landscaping. When someone (me, or the kids) busts a hole in a wall by accident, I think "Dangit, I have to fix that..." rather than "well, there goes my security deposit".
I'm sure that a big part of the appeal of home ownership is that our parents valued it, but I feel like there's a real intrinsic value to it as well. Consider surveying your friends who have owned + also rented, and ask them about the things they like or dislike about it.