The speaker is contemplating the picture on a Grecian urn that shows young people leaving a city for a spring religious festival. They are forever young, forever happy, forever about to fall in love, forever enjoying life on a beautiful day. As the speaker thinks about these unchanging figures, he emphasizes their immortality to draw a contrast between it and human mortality.
The speaker identifies with the scene on the urn to the point of envy, wishing that like art, life could freeze in time at a high point of youth and happiness. That way, nobody would have to suffer, age, or die. Art seems superior to human life to the speaker precisely because it is not time bound but instead captures beautiful moments forever.
In the third stanza, the exact center of the poem, the speaker hits a high point of ecstasy as he contemplates the urn's immortality, repeating the word happy over and over. He begins the stanza by writing,
Ah, happy, happy boughs! that cannot shed
Your leaves, nor ever bid the Spring adieu.
In other words, he feels how delightful it would be to be in a spot that is forever spring, where the boughs of the trees always have green leaves and he would never have to say goodbye to spring's warm weather.
This tension between art's freezing of time and the human experience of passing through time is a common theme in poetry. Being frozen means stasis, no change or growth; but it also means, if you freeze life at the right moment, no suffering, aging, or death.