Learn Turkish: Bu sırrı kimseye söylememek üzere seninle paylaşıyorum. - I am sharing this secret with you on the condition that you do not tell anyone.

You are right that -mAk üzere often means "about to" (like being about to leave). However, as this lesson shows, it is also used to express a strong condition or proviso, meaning "on the condition that" or "provided that". Here, it attaches to the negative infinitive söylememek (not to tell), creating the meaning "on the condition of not telling".

Why does "anyone" translate to kimseye? Doesn't kimse mean "nobody"?

Kimse actually just means "anyone" or "anybody". To get the meaning of "nobody", Turkish pairs kimse with a negative verb. Since the infinitive here is negative (söylememek), the combination of kimse and the negative verb gives us "not to tell anyone" (or "to tell nobody").

Why is there a double 'r' in sırrı?

The root word for "secret" is sır. When we add a suffix starting with a vowel, like the specific direct object (accusative) suffix -ı in this sentence, the final consonant doubles. This is a common rule for several short words borrowed from Arabic, like hak becoming hakkı.

Why does kimseye take the dative case ending -e instead of the accusative?