That's how I tend to motivate things, not really a concrete application, just shifting focus...
The local Galois group is a profinite, hence compact group. Picturing this group is difficult. Two ways are known:
- Understanding it in terms of generators and relations
- Understanding it in terms of its representation category (Tannaka-Krein)
Classification 1 is known in some cases, 2 is probably difficult, so having a different "equivalent" category to work with seems desirable.
Now, if somebody ask for a motivation of the Galois group, you answer: "Are you kidding me?";)