Learn Danish: Jeg har købt en elektrisk tandbørste. - I have bought an electric toothbrush.

Because it describes tandbørste, which is a singular en word (common gender). In Danish, an adjective gets no special ending when describing a singular, indefinite en word. If the noun were plural or definite (like "the electric toothbrush"), the adjective would get an -e ending: den elektriske tandbørste.

How can I tell that tandbørste takes the article en instead of et?

Tandbørste is a compound word made of tand (tooth) and børste (brush). In Danish, compound nouns always take the gender of their final word. Because it is en børste, the combined word is also an en word (en tandbørste).

I often mix up købt and købte. Why do we use købt in this sentence?

Købt is the past participle of the verb at købe (to buy), and it is used here together with the helper verb har to form the present perfect tense (har købt = have bought). Købte is the simple past tense (bought). You would only use købte if you dropped har, like in: Jeg købte en elektrisk tandbørste (I bought an electric toothbrush).