I am fairly inexperienced with elliptic curves so there might be aspects of my question that may need better wording but let me know if there are any issues:
Question: Say I have an elliptic curve over $\mathbb{F}_7$ and this curve has 12 points. I take a subgroup of size 3 and I quotient the curve by that subgroup. Magma and Sage can easily tell me the equation of the curve where the quotient lives. Not surprisingly, extra points pop up when taking a quotient that where not defined over $\mathbb{F}_7$ but become defined over $\mathbb{F}_7$ when you take the quotient. So the curve they spit out may (and usually does according to my random sample) still have 12 points.
What is happening on the function field side? Is the function field of the original curve an extension of the function field of the other curve? If not what is going on?
Such a map is called an isogeny, and isogenies over finite fields preserve the number of points (in fact, two elliptic curves over a finite field are isogenous iff they have the same number of points).
On the function field side, this corresponds to an embedding of the function field of the original curve into the second one. By the existence of the dual isogeny, you also have an embedding of the second into the first. All of this is explained quite nicely in the first few chapters of Silverman's Arithmetic of Elliptic Curves book.