The effect that this has on the meaning of the poem is this: it hints at the fact that the speaker might be artfully concealing the complete truth.
The reason for the line breaks
The artful placement of line breaks in this poem can create a sense of uncertainty and ambiguity, which suggests that the speaker might not be revealing the whole truth.
This is because the pauses caused by the line breaks allow the reader to pause and reflect on what has been said, and to consider whether the speaker is being entirely honest. So, the idea is that the speaker is hoarding the truth.