There is no word for 'the' in the original Czech, and (without further context) it is incorrect to add it in the English translation. Only if the Czech sentence included a demonstrative, such as some form of 'to', would it be correct to use 'the' or 'that' in the English. (That is the principle that this DL course follows, in any case.)
To put the matter another way, the Czech sentence here is not about particular cats and particular birds. Rather it is a general statement about all cats and any birds.
Kočky žerou ptáci. means Birds eat cats. because ptáci is in nominative and kočky in accusative.
You should really review the Tips and notes about Czech pronunciation and orthography. The two words are written differently because they are pronounced differently.
ptáci - [ptáci]
ptáky - [ptáki]
Kočky is the same in nominative and accusative: [kočki]