Learn Swedish: Det finns en stig genom ängen. - There is a path through the meadow.

The preposition genom translates to through. Just like in English, you use it for a path that goes from one side of an area to the other, especially when the area has some sort of volume, like the tall grass of a meadow.

Could I use över instead of genom here?

Yes, you could say en stig över ängen meaning a path across the meadow. Using genom paints a slightly different picture, implying the meadow is full of tall grass that the path cuts directly through, rather than just crossing a flat surface.

Why do we use Det finns for there is instead of literally translating it to Där är?

The phrase Det finns is the standard Swedish expression for there is or there are when talking about the general existence of something. If you used Där är, it would specifically mean over there is, as if you were physically pointing at the path.

What is the base form of ängen?

The base form is en äng which means a meadow. Because it is an en-word, we add the suffix -en to make it definite, creating ängen for the meadow.