Translation:The cats were sleeping the day before yesterday.
It depends, I think, on the formality of the conversation; dropping the "the" is just relaxed speech, assumed. I might say to a banker I did not know that "I opened my account the day before yesterday", but to my friend who asks when you left a voicemail on their phone, "Day before yesterday. "
Hindi is one of a couple (few?) of languages descending from Sanskrit that have a word for "the day before yesterday"/"the day after tomorrow".
- कल = "yesterday", when in a statement about the past
- कल = "tomorrow", when in a statement about the future
- परसों = "day before yesterday" if you're talking about the past
- परसों = "day after tomorrow" if you're talking about the future