Any spanning tree of a graph with |V| vertices has how many edges?

A tree on n vertices has exactly n − 1 edges. Any more would create a cycle (violating tree-ness) any fewer would disconnect the graph (violating spanning). So a spanning tree of |V| vertices has |V| − 1 edges. The expression |E| − |V| + 1 is the cyclomatic complexity / circuit rank, not the tree size.