Every path from the top of the diagram to the bottom is one term. Collect them all and you have the chain rule.
Click a route to collect it. The chain rule is finished when you have them all.
Click a path to collect its term.
When one quantity depends on others that depend on others in turn, the chain rule tells you how a nudge at the bottom reaches the top. The tree makes it mechanical: every route from top to bottom is one term, and the term is the product of the derivatives along it. Add the routes together and you're done.
That's the entire rule. In one variable there is only ever one route, so the sum has one term and the rule looks like a single product — which is why the multivariable version can seem like a different theorem. It isn't; there are just more ways down.
The sum is doing something specific. Nudging t changes x and changes y, and both of those changes push on z independently. Each contribution has to be counted, so the routes add. Along a single route the derivatives multiply instead, because there the changes are passed along in series — a nudge in t becomes a nudge in x becomes a nudge in z.
Once you're reading the tree, the notation stops mattering. Draw the dependencies, count the routes, and you can write down chain rules for arrangements you've never seen before without remembering any formula.