Lord Tennyson's poem "Ulysses" is written from the perspective of the hero whose name appears in the title of the poem. Ulysses is his Roman name; the Greeks called him Odysseus and the most extensive account of his adventures appears in Homer's Odyssey, which was written around 725 BCE.
Whereas Homer's Odyssey focuses on how Odysseus managed to make it back home to Ithaca after ten years of fighting at Troy, Tennyson's poem shows the hero looking back after his adventures and after his return to Ithaca.
The phrase "hungry heart" appears in the context of the larger phrase at lines 12-13 of the poem: "For always roaming with a hungry heart / Much have I seen and known". The reference here is to Odysseus/Ulysses and his various wanderings around the Mediterranean region after the fall of Troy.
I would say that the phrase "hungry heart" refers to Odysseus' insatiable curiosity about the different peoples that he encountered. He had himself lashed to the mast of his ship and he stuffed his crew's ears with wax so that he could listen to the song of the deadly Sirens. He ended up getting several of his men killed in his curiosity about the Cyclops in Odyssey 9. Odysseus' heart was, indeed, hungry, hungry for adventure and hungry for the glory that encounters with strange and unusual creatures and peoples might bring to him.