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:

1. Understanding it in terms of generators and relations
2. 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?";)

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:

1. Understanding it in terms of generators and relations
2. 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, you have reduced the question why we should bother about the local Galois group (see discussion in the comments).