Learn Dutch: Ik speel buiten met de kat. - I play outside with the cat.

This is how the vowel dropping rule works when we make verbs shorter! The full verb is spelen. If you split it into syllables, you get spe-len. The first syllable is open (ends in a vowel), so one e is enough to make a long sound. But to make the ik form, you drop the -en, leaving spel. Because spel ends in a consonant, the syllable is now closed. To keep that same long vowel sound, we must add an extra e, making it speel.

Does buiten mean out or outside?

It means outside or outdoors. You might remember the word uit for out, but when you are talking about being physically outside in the open air, you always use buiten.

Is there a rule for why it is de kat instead of het kat?

In Dutch, nouns use either de or het for the word the, and you mostly just have to memorize them. However, almost all animals are de words. You will see de hond (the dog) and de muis (the mouse), with only a few exceptions like het paard (the horse).

Could I say Ik speel met de kat buiten like in English?