Firstly, I would like to understand more about "spanning tree". I have read some definition, but still, cant 100% understand. It would be better if there is a graph to illustrate.
Secondly, comes to the main question. Why is the number of edges of the spanning tree (T) = V-1? (V: vertices) Some references use this as a fact but I can't find why.
Thank you very much.
You can find graphs to illustrate the concept of a spanning tree on Wikipedia.
That a spanning tree has $V-1$ edges can be proved by induction: For the base case, a spanning tree for $1$ vertex has $0$ edges. For the induction step, delete any edge. This cuts the spanning tree into two spanning trees for two vertex subsets with, say, $V_1$ and $V_2$ vertices, with $V_1+V_2=V$. By the induction hypothesis, these have $V_1-1$ and $V_2-1$ edges, respectively. Including the deleted edge, this yields a total of $(V_1-1)+(V_2-1)+1=V_1+V_2-1=V-1$ edges in the original spanning tree.