You may know the word "drink" is to take liquid into your mouth and swallow. Drink water, drink juice, drink wine. The phrasal verb to "drink up" means to drink all of something completely.
There are usually three ways we can use this. To drink up or drink something up or drink up something.
In the present simple we use drink up or drinking up in the -ing form. The past simple of this phrasal verb is drank up (The milk was going bad so I drank it all up).
EAT UP is another phrasal word that's very similar. Instead of drinking something completely, we eat something completely.
"Eat up! This is your last meal of the day!"
"Eat your vegetables up!"
If you'd like to look at some other common phrasal verbs in English, the English Club has a great table here.