I implemented Dijkstra using Fibonacci heaps a few years ago, and the problem is pretty similar. Basically, the advantage of Fibonacci heaps is that it makes finding the minimum of a set a constant operation; so that's very appropriate for Prim and Dijkstra, where at each step you have to perform this operation.
Why it's good
The complexity of those algorithms using a binomial heap (which is the more "standard" way) is O(E * log V), because - roughly - you will try every edge (E), and for each of them you will either add the new vertex to your binomial heap (log V) or decrease its key (log V), and then have to find the minimum of your heap (another log V).
Instead, when you use a Fibonacci heap the cost of inserting a vertex or decreasing its key in your heap is constant so you only have a O(E) for that. BUT deleting a vertex is O(log V), so since in the end every vertex will be removed that adds a O(V * log V), for a total O(E + V * log V).
So if your graph is dense enough (eg E >> V), using a Fibonacci heap is better than a binomial heap.
How to
The idea is thus to use the Fibonacci heap to store all the vertices accessible from the subtree you already built, indexed by the weight of the smallest edge leading to it. If you understood the implementation or Prim's algorithm with using another data structure, there is no real difficulty in using a Fibonacci heap instead - just use the insert and deletemin methods of the heap as you would normally, and use the decreasekey method to update a vertex when you release an edge leading to it.
The only hard part is to implement the actual Fibonacci heap.
I can't give you all the implementation details here (that would take pages), but when I did mine I relied heavily on Introduction to algorithms (Cormen et al). If you don't have it yet but are interested in algorithms I highly recommend that you get a copy of it! It's language agnostic, and it provides detailed explanations about all the standards algorithms, as well as their proofs, and will definitely boost your knowledge and ability to use all of them, and design and prove new ones. This PDF (from the Wikipedia page you linked) provides some of the implementation details, but it's definitely not as clear as Introduction to algorithms.
I have a report and a presentation I wrote after doing that, that explain a bit how to proceed (for Dijkstra - see the end of the ppt for the Fibonacci heap functions in pseudo-code) but it's all in French... and my code is in Caml (and French) so I'm not sure if that helps!!! And if you can understand something of it please be indulgent, I was just starting programming so my coding skills were pretty poor at the time...