'I think. Thinking things exist. Therefore, I exist as a thinking thing.' This argument is

The argument is valid because its conclusion follows necessarily from its premises.

Premise one asserts 'I think', premise two that thinking things exist.

If I think, then I am a thing that thinks, hence a thinking thing.

Since thinking things exist, my existence as a thinking thing follows.

The conclusion cannot be false while both premises are true, which is validity.