There's a tradition that the Spartans descend from the Herakleidae, the descendants of Herakles, and the Dorians. It's one that Tyrtaeus references in other poems, so it's certainly not too much of a leap to think he's doing the same thing in fragment 11.

Tyrtaeus fragment 19 (I'm using the Loeb numbering, in case there's any confusion) references men of the three Dorian tribes said to be descended from Hyllas taking part in a battle. The military context of the reference, used to identify the men lining up in formation, makes it a nice parallel to 11's exhortation. Fragment 2 likewise references the Spartan descent from Herakles, this time not through Hyllas but specifically as the Herakleidae, in one of the earliest extant references to the identification of the Dorians as Herakleidae we have. The ties to the Spartan state in fragment 2 are the clear focus, as the poem says that:

the Kronus-descended husband of beautiful-crowned Hera, Zeus himself, gave this town to the Herakleidae, leaving together with them windy Erineus, we came to the wide island of bright-eyed Pelops...

So fragment 11 is one of at least three named references to the Herakleian ancestry of the Spartans within a very fragmentary extant corpus for Tyrtaeus, if that gives a sense of how central a figure Herakles was to Spartan identity in Tyrtaeus' poetry.

If you're interested in ancient Sparta, you may want to check out:

Malkin, Irad. Myth and Territory in the Spartan Mediterranean. Cambridge: Cambridge University Press, 1994.