Learn German: Ich bin zu Hause.

You would use Wo (where) because this sentence describes a fixed location. The question would be Wo bist du? (Where are you?). In contrast, you use Wohin (where to) for movement toward a destination, and Woher (where from) for movement away from a place.

Doesn't zu usually mean 'to'? Why does it mean 'at' here?

Yes, zu usually means 'to', but zu Hause is a special fixed expression that means 'at home'. If you want to say you are going 'to' your home, which shows movement, you use a different fixed phrase: nach Hause.

Why is there an 'e' at the end of Hause instead of just Haus?

In older versions of German, an 'e' was often added to the end of nouns after certain prepositions. While modern German dropped this rule, the extra 'e' survived in a few very common everyday phrases like zu Hause (at home) and nach Hause (going home).

Why is Hause capitalized in this phrase?

In German, every single noun is capitalized. Because Hause is a form of the noun Haus (house or home), it gets a capital letter even when it is part of a fixed expression.