Learn Spanish (Spain): Estoy barriendo el suelo de la cocina.

It comes from the verb barrer (to sweep). To say you are doing an action right now, you drop the -er ending and add -iendo. This gives you the "-ing" form (the gerund) for -er and -ir verbs.

Why is it el suelo de la cocina instead of something like "la cocina suelo"?

In English, you can put two nouns together like "kitchen floor". Spanish doesn't do this. Instead, you have to use de (of) and say "the floor of the kitchen": el suelo de la cocina.

Why do we use estoy instead of soy for "I am" in this sentence?

Whenever you are talking about an action that is happening right now (using the present continuous tense), Spanish always uses the verb estar, never ser.

Do I need to include yo and say "Yo estoy barriendo"?

No, you don't need it. The verb form estoy already tells us exactly who is doing the sweeping ("I"). In Spanish, it is very common to leave out pronouns like yo unless you really want to emphasize who is doing the action.