Learn Finnish: Uusi ohjelma auttaa markkinoinnin ulkoistamisessa. - The new program helps in outsourcing the marketing.

Because ulkoistaminen acts as a noun here (a verbal noun). In Finnish, the object of a -minen noun takes the genitive case. You can think of it literally as "the outsourcing of the marketing."

The lesson mentions that activity verbs can take either the illative (-miseen) or inessive (-misessa). Why does auttaa take the inessive here?

The verb auttaa usually takes the inessive case when combined with a noun or a -minen word. It expresses being involved within a situation or process. You are helping in or with the outsourcing (ulkoistamisessa).

Could we just use the 3rd infinitive and say "auttaa ulkoistamaan"?

Yes, you could say Uusi ohjelma auttaa ulkoistamaan markkinoinnin. Both are grammatically correct, but they have a slightly different feel. Using the 4th infinitive in the inessive (ulkoistamisessa) emphasizes helping with the ongoing process or the general task, while the 3rd infinitive illative (ulkoistamaan) focuses more on achieving the specific goal.

What is the root of the word ulkoistamisessa?

It comes from the verb ulkoistaa (to outsource). This verb is cleverly derived from the word ulko- (outer, external). Adding the -istaa suffix turns it into a causative verb meaning "to make external," which is exactly what outsourcing is!