Let me just say that there is a very good reason that people make "Oh, THIS will be the Year of the Linux Desktop" jokes year after year. The Linux desktop ecosystem feels flakier to me now than it did in 2005. There are too many geeky alpha males vying for dominance right now for me to recommend that option to anyone (sysvinit vs systemd, Xorg vs Wayland vs Mir, and both of those are absolutely HUGE parts of the user experience and OS). If all you want to do is browse the web, grab a Chromebook (Linux-based, but it's a very tightly designed system). If not, and you still want to move away from Windows, look into getting a Mac.
If you're trying to do anything from within Windows 10 itself it's definitely not going to work. You can't format the OS drive in any version of Windows while the system is running since that would lead to Critical Existence Failure of the environment that's currently loaded. You also won't be able to initiate the Windows 7 install from within Windows 10, as in-place downgrades aren't supported (and even if they are, they'd be a very bad idea--I'm even skeptical about in-place upgrades).
That certainly doesn't mean that Windows 10 is preventing you from installing an operating system. It's a safety measure. Trying to install an older version of Windows over a newer one can cause a ton of problems. There are a lot of subtle things that can change regarding the disk layout and filesystem that can cause the older, outdated Windows 7 tools to not work properly against newer installations.
To install an older system, your best bet is to boot directly from disc, clear the system of all partitions (not just formatting, as that will keep the newer, likely incompatible boot and recovery partitions intact) and reinstall from scratch.
First and foremost, make a backup of any data that you care about.
Next, to actually wipe the partition table and more or less start from scratch, you'll have to boot from the Windows 7 media. Your system will probably have a key, such as F12, that you can press to get a boot menu. Failing that, you can rearrange the drive order in the BIOS config so that your optical drive is above your SSD in the boot process. When the disc starts to boot, tap a key to boot from the disc.
When the setup program loads, do a custom install. When you get to the section where you configure your partitions, click on the "Drive Options (Advanced)" text. This will add some additional buttons that allow you to add and remove partitions. Click on each partition on your system disk, click Delete, and accept the prompt. If you're at all uncomfortable about this step and are worried about deleting the wrong ones, power your system off and disconnect your other hard drives or SSDs.
When you're done, the only item that should be left for that specific disk is "Unallocated Space." As long as that is selected, Windows will take care of everything else when you click the Next button. You won't have to create or format the partition manually. After that's finished, you'll have a fresh drive and a new Windows 7 install.
One thing to bear in mind is that Windows 7 is no longer receiving mainstream support--it's currently in an extended support window until January 14, 2020. It will continue to receive security updates during this time, but feature updates are not guaranteed.