This passage occurs very early in the novel. In it, we are primarily inside of Clarissa Dalloway's head as she embarks on shopping for her party.
The writing is largely stream-of-consciousness, meaning we are overhearing Mrs. Dalloway's thoughts as she has them. But Woolf always makes sure the thoughts people experience occur in the context of physical reality. In this case, Mrs. Dalloway is noting how quiet it is in the park. She sees the mist, the ducks, the waddling bird, and the government buildings in the background. These all become a backdrop as she notices her old friend Hugh Whitbread approaching, carrying a box "stamped with the Royal Arms." This box shows us from the start that he is somehow connected with the royal court. Mrs. Dalloway thinks of him as the "admirable Hugh," a positive assessment. Her thoughts go from noticing what he carries to a quick judgement about his character.
At this point, they have a rather banal dialogue. Hugh asks her where she is going, and Clarissa evades a direct answer by saying she loves walking in London.
Hugh doesn't pursue where Clarissa is going, suggesting he's not really interested—he really wants to tell her what he is up to. We receive a summary of what he says to his old friend rather than a record of his speech: he and Evelyn have come to London to consult doctors about Evelyn's "internal ailment." He also tells her he may be late to her party.
As he talks, we are also privy to Clarissa's thoughts: Hugh makes her, for example, feel inadequate about her hat. Although she likes him and has positive thoughts about him, the scene also sows our doubts about him. He is condescending and perhaps a bit phony to Clarissa, trying to flatter her by saying she looks like a girl of 18 (as if this is what matters to a woman) and showing off his importance by saying he might be late to her party because of another obligation. We also find out that Clarissa's husband is driven crazy by him and that Peter Walsh has never forgiven her for liking him.
The passage establishes motifs that will recur throughout the novel: illness, Clarissa's party, the pretensions of patriarchal men like Hugh, and the reappearance of Clarissa's old flame Peter Walsh, back from England.
While there is description and dialogue, the passage helps establish Woolf's contention that nine-tenths of what goes on within a person is hidden. Hugh and Clarissa have a quick dialogue, but Clarissa's internal, stream-of-consciousness thoughts tell us much more than anything either she or Hugh says. More importantly, what we know about Hugh comes from Clarissa: there is no omniscient narrator to tell us if her perceptions are right or wrong. Woolf only offers us Clarissa's subjective impressions.