Mac Freeze and Screen Goes Weird After RAM Upgrade
TL;DR: You probably want to exchange the RAM you bought for new sticks.

From your picture, it looks like that's a MacBook Pro (13-inch, Mid 2012). As you mentioned, it does support 2x8 GB (16 GB total) at 1600 MHz (PC3-12800).

If the issue doesn't occur with the original RAM, you can be fairly sure that your memory slots are good. It's very likely that the issue lies in the modules themselves.

You can use a tool like Memtest86+ to test the RAM completely to help narrow down where the issue is. It could be one stick, or it could be both.

I recommend installing a single 8 GB stick, running through the full battery of tests, and then repeating with the other stick in the same slot. By testing them one at a time, you can easily determine which module is bad.

If both pass, then you can try the same together. If they only fail as a pair, there could be a timing issue or something else going on with the RAM.

To be absolutely sure that there's not an issue with your memory slots (read: logic board), you should test your old modules as well.

Finally, you mention that you purchased 1600 MHz RAM, but you'll want to confirm that. A common issue with Mac RAM installation is the use of higher clocked RAM under the assumption that it will correctly work on boards that want a lower clock.

For example, if you bought 2x8 GB at 1833 MHz, you might expect it to downclock to 1600 MHz for your computer. While normally that's true, Mac logic boards are very sensitive to the clock speed. Often, you'll get kernel panics, graphical issues, and general instability even though it may report as the clock speed you expect.

The solution is to purchase RAM that's the exact speed you want.