Learn German: Gute Nacht! Ich schlafe jetzt. - Good night! I am going to sleep now.

Greetings like these are actually shortened sentences! You are secretly saying 'I wish you a good night.' Because Nacht (night) is a feminine noun, the adjective gut gets the feminine ending -e. Morgen (morning) and Tag (day) are masculine, so they get the masculine ending -en.

The translation says 'I am going to sleep,' but doesn't ich schlafe just mean 'I sleep'?

Yes, it literally means 'I sleep.' German doesn't have a separate '-ing' form like English (I am sleeping). You just use the normal present tense. By adding jetzt (now), it naturally translates to 'I am sleeping now' or 'I am going to sleep now.'

Is it wrong to translate 'I am going to sleep' literally as Ich gehe zu schlafen?

Yes, that would sound very strange to a German speaker! If you want to emphasize the physical action of going to bed, you would say Ich gehe jetzt schlafen (I go now sleep) without a word for 'to'. However, simply saying Ich schlafe jetzt is perfectly normal and common.

Why does the verb schlafe end in an -e?

In German, verb endings change depending on who is doing the action. The dictionary form of the verb is schlafen (to sleep). When the subject is ich (I), you drop the -en from the end and add an -e, making it ich schlafe.