What you’re being offered there is the easiest and most recommended way.
It’s offering you updates to kernels that are already installed.
It’s very safe and it’s easy to switch if things go wrong: just press and hold SHIFT on bootup, select “Advanced options for…” and select the kernel.
Framework hardware is so new that generally you’ll want the most advanced kernel on offer (depending on which hardware generation you have). You can do this through the window you posted - View - Linux Kernels.
Yes, I very much like Mint update manager. Great to know then, I will go ahead. I was just hesitant as I remember after setting up my system I had to install a very specific kernel and was worried that perhaps I am not supposed to change that.
This is probably a minimum kernel, i.e. the oldest one that will still run on the hardware. That support is built-in moving forward.
It depends on your hardware, but that 5.15 kernel will probably run OK on 11th gen, but for 12th gen, 13th gen and AMD you probably want at least the 6.5 kernel. There’s now a 6.8 kernel as well.