Learn Russian: Горло красное.

The noun горло (throat) ends in -о, which means it is a neuter noun. Adjectives must match the gender of the noun they describe, so we use the neuter adjective ending -ое instead of the masculine -ый or feminine -ая.

Where is the verb "is" in this sentence?

In Russian, the present tense of the verb "to be" (am/is/are) is almost always skipped. You simply place the subject (горло) right next to its description (красное) to say "The throat is red."

Is this a common phrase to hear at the doctor?

Yes, very common! While English speakers usually focus on the pain and say "I have a sore throat," a Russian doctor looking in your mouth will often simply state Горло красное (The throat is red) as a standard way to confirm you have a throat infection.