You will undoubtedly receive a variety of different answers to this question, as one's reading of poetry is essentially a personal affair. For me, one of the key phrases in this poem is the description Ulysses gives us of how experience can never be satisfied or exhausted:
Yet all experience is an arch wherethrough
Gleams that untraveled world whose margin fades
Forever and forever when I move.
This of course captures the central theme of this poem, which is the way in which Ulysses is not able to accept a quiet life and is determined to live his life to the full, making the most of every single second he is given to gain new experiences and not become stagnant. The metaphor in this quote is a beautiful rendition of how each new experience we gain only leads us on to gain yet more new experiences, and how we are unable to ever reach the limit or end of experience. It points towards a true questing spirit that can never be happy sitting idle, and is a model to all of us.
I think that the closing line is the key line in the whole poem and critical to unlocking both its meaning and greatness. Honestly, there are many great lines in the poem, but the closing is amazing:
To strive, to seek, to find, and not to yield.
On one hand, this line can be a testament to the greatness of the poem's subject. Ulysses can be seen as a permanent warrior, someone who will not be withered with time. His desire to continue the pursuit of his life's work is evident in these words. Tennyson's construction of Ulysses is one who will always live in accordance to his spirit and there is a beauty in that. In a world where so much is fleeting and subject to contingency, Ulysses stands as a benchmark of this lack of change. Even though he has safely returned home, he will not use that as an opportunity to deny the nourishment for adventure that his soul obviously seeks. It is here where this line reflects much that is great about its subject.
Simultaneously, it is here where some of the worst elements of Ulysses are seen. The fact that Ulysses cannot honor in action the love and devotion of Penelope is startling. There can be fewer visions of virtue and loyalty than Penelope to Ulysses. She pined and suffered for his return. The gratitude that she gets is a condemnation of the domestic life she leads and envisions sharing with him, her husband. After everything, there is a level of selfishness revealed in his declaration of the last line. At the same time, the last line does not speak to how important the son's relationship to his father is. Telemachus wept for his father, and never wavered in his commitment to see his father return. While political provisions are made for Telemachus, there is little in way of an emotional farewell in the idea of "To strive, to seek, to find, and not to yield." In seeing a father tell his son that he is, for all practical purposes, an impediment to his dreams is repugnant.
In the end, this is why this line is so important. It reflects so much in Ulysses' character. At the same time, this line helps to bring out a rather complex rendering of Ulysses, one that compels the reader to think and reflect, creating difficulty in judgment. This becomes the testament to the line, the poem, and its place in literature.