“To say aloud, to tell” (Shijing) > “to recite” > “to read”.
Cognate with Tibetanཀློག(klog, “to read; to recite”). Starostin sets up Proto-Sino-Tibetan*k-lok(“to recite; to give notice”) for this, and also includes Burmeseလျှောက်(hlyauk, “to tell; to ask; to petition; word-for-word translation; indiscriminately”) and Mizothlûk(“accent, tone, intonation”). Related to 讟 (OC *l'oːɡ, “resentment, slander < to grumble; complaint”).
Sagart (1999) argues that “read” cannot be the primary meaning, since existence of the concept “to read” before the invention of writing is questionable. There is likely no writing at the time of the split between Chinese and other Sino-Tibetan languages, therefore the Tibetan word may represent a borrowing from Chinese.