iPhone 5S keeps switching from blue screen to Apple logo

Hello,

So my iPhone has been recently (starting last night) began to restart and have a blue screen. It happened while I was trying to change my music but I am still freaked out about it since I plugged it in and it did the same thing again. At one point before it got bad, I was able to see my messages and I realized it was showing me my old ones. I was confused since I just sent a few messages and got some replies after I sent them. I told my mom I want a new phone but Verizon says you need $200 I think. I dunno.

I haven't tried to fix it since I figured you couldn't lol.

I have the newest iOS update on it. I believe that is like 9.1.0?

It's also 16 gb storage.

If I missed anything, just comment below. Thanks!