The poem's opening lines are literally dramatic: quoting the speaker's landlady, they plunge the reader straight into a miniature drama, complete with characters, dialogue, setting, and exposition. There's something ironic about this theatrical beginning. As the poem goes on, it will focus on how undramatic the setting here is—how little happens.
The person quoted is a landlady, who's showing the speaker a room that she's renting out and telling him about the previous tenant. Evidently, she got to know that tenant fairly well over the years—but the two probably weren't that close, as she still refers to him by the formal "Mr Bleaney."
There's something faintly comic about this name. It echoes homely words like "beans" and "bleary"; shares its initial /bl/ sound with words that connote banality, like "bland" and "blah"; and rhymes with colloquial words for smallness, such as "teeny." Bleaney already sounds like an average, forgettable old chap.
Bleaney rented this room (likely part of a bedsit or cheap boarding house) "the whole time he was at the Bodies"—a term that the poem doesn't explain, but that clearly refers to his workplace. (Larkin once suggested that "the Bodies" was an auto works; there was also a taxi manufacturer called Carbodies in his hometown of Coventry.) But the nickname "Bodies" inescapably calls to mind human bodies, like the depersonalized bodies of workers at a grueling job. It also calls to mind dead bodies, especially in the context of the rest of the landlady's sentence:
[...] He stayed
The whole time he was at the Bodies, till
They moved him.’ [...]
"They moved him" could mean that his employer transferred him elsewhere, but it could also be a euphemism implying that Bleaney died and was carried out of the room. (Remember that the landlady is trying to rent the room; she might not want to drive the speaker away by mentioning death outright.) Either way, her remark invites thoughts of death and makes the room seem a little haunted.
Both of the first two lines are enjambed. The enjambment after line 1 emphasizes the word "stayed" and makes it linger—stay—in the reader's mind for a moment. Until the sentence continues, it reads simply "He stayed." In a way, that simple phrase seems true of Bleaney: his ghostly presence seems to linger in the room.
The enjambment after line 2 creates a pause after "till." If the landlady is hinting that Bleaney died here, the pause might reflect her awkwardness as she decides how to share this fact.