Essentially the same as Mason and Mike have already said, but in different words...
If you have no edges, the most (connected) vertices you can have is one. Call this vertex the root.
To add a new vertex to a tree, you must also add a new edge (in order that the new vertex is connected). That edge can (and must) connect to any pre-existing vertex. The new vertex is a child of the pre-existing vertex, and we don't limit the number of children a parent can have.
If you start with just the root (1 vertex and 0 edges) you end up with 2 vertices and 1 edge. Repeat n times to get n+1 vertices and n edges.
This isn't a full proof in itself, but it's impossible to add edges or vertices in any other way (except that you could add two children at once, provided that you do it in a way that's equivalent to doing adding one first then the other). You can't add an edge without adding a vertex because doing so would complete a cycle. You can't add a vertex without adding an edge because that vertex wouldn't be connected to the tree.
BTW - for undirected graphs, words like "root", "parent" and "child" don't mean much, at least not formally. Any vertex in any undirected tree can be considered the root, and which vertex you happen to call the root decides for all edges which vertex is the parent and which is the child. My mental image of a tree tends to include identifying a particular vertex as the root, but that's arguably a misleading mental image.