Learn Russian: Это здание выше нашего дома.

There isn't one! In Russian, you can compare two things by putting the second thing in the Genitive case. Since we are comparing the building to "our house", наш дом becomes the Genitive нашего дома to mean "than our house".

Could I use чем (than) to write this exact same sentence?

Yes, you could say Это здание выше, чем наш дом. When you use чем, the second item stays in the Nominative case (наш дом). But dropping чем and using the Genitive case, as shown in this sentence, is very common and natural in Russian.

What is the base dictionary form of выше?

It comes from the adjective высокий (high or tall). Like many common Russian adjectives, its comparative form is irregular. Instead of a standard ending, it changes to выше. Notice that short comparative forms like this don't change for gender, number, or case.

Why does the sentence use Это здание instead of Этот здание?

The word здание (building) is a neuter noun because it ends in -е. Therefore, the demonstrative pronoun for "this" must also be in its neuter form, which is Это.