My kitchen floor stains quickly, when i splash water on it - help

was it sealed

You need to seal it. https://www.youtube.com/watch?v=c3pWFfwwoFY

You can seal the tiles. It may or may not change the look of them but you won't see water marks on them.

When the water dries what happens?

I would recommend getting a water repelling coating for your floor. Don't buy the cheapest one out there though, because they can end up making your floor look weird.

I learned a good trick for my vinyl kitchen floor for cleaning. I use a 3 part mix of water, baby oil and white vinegar. Let dry and lightly wet mop when needed. It works great for me, I have seen many other ideas out there so try it and see the results. I don't think it should hurt the finish. You might want to check with the store you purchased from just to be safe.

http://homeguides.sfgate.com/apply-sealant-slate-tile-37161.html