A "puddle" is a small area of water. Puddles can form after it rains, when snow melts, or even in your house if one of your water pipes breaks.
When you "step in" a puddle on a road or sidewalk, it means that you put your foot into the water.
(do something) on the way over (somewhere)
This means to do something while you're traveling to a place:
I ran into Jeff Skillings on the way over here. You remember Jeff, right?
Sometimes you can leave out "over":