You could call Kenya and Tanzania the most noteworthy Swahili-speaking countries, yes. Those are also the two countries where the course creators are from.

Here's a map of the language area:

Tanzania (1) is the only country that has Swahili as its only official language. In Kenya (2) and Uganda (3) it shares that status with English.
Rwanda (4) recently added Swahili as its fourth official language (after Kinyarwanda, English, and French). Swahili is also one of the official languages of Congo (6), although it's only widely spoken in the eastern part.

Swahili is also widely spoken in some neighboring areas (light green), most notably Burundi (5), which is a relatively small country right in the middle of the Swahili language area.

The thin dark areas among the coasts is where the Swahili tribes live, the people who the language originally belongs to. They are the reason why the name "Swahili" comes from an Arab word for "coasts".

Mainly, I think it's also spoken in the eastern part of the Democratic Republic of the Congo. I also saw that they were spoken a little in Burundi and Rwanda?

And Uganda, I knew I forgot one.