Learn Russian: При чтении статьи без интернета обычно экономишь зарядку. - When reading an article offline, you usually save battery.

Using при + the Prepositional case of a verbal noun (like чтении from чтение) is a very common way to express the circumstances or conditions under which something happens. It means "during the reading" or "in the condition of reading." It is slightly more concise than saying когда читаешь and neatly sets up the prerequisite for the rest of the sentence.

Why is статьи in the Genitive case rather than the Accusative?

Because it directly follows the noun чтении, not a verb. While you read an article (читаешь статью - Accusative), when you turn the action into a verbal noun, its object takes the Genitive case to show relation. Think of it literally as "the reading of an article."

Why is the verb экономишь in the "ты" (you) form, but the pronoun is missing?

This is a "generalized personal sentence." Just like the generic "you" or "one" in English, Russian drops the pronoun ты and uses the 2nd person singular verb to state a general rule or common experience that applies to anyone, not just the specific person you are talking to.

Doesn't зарядка mean "charger" or "morning exercises"? Why is it translated as "battery" here?