What is the construction of this part?

Question

Noise, when benign in nature, much conduces to the amelioration of spirits. In a coffee house, say, what may explain that buoy in the bosom, that warmth, that sense of security, yea, assurance, of one's not being an island as we say, but that higgledy-piggledy gabble in the surrounds of one's solitary table which may well otherwise give rise to aggravation, blessing one with an illusory sense of company? The choice of place matched with one's purpose is essential; contemplation by a babbling brook, in a moderately bustling cafe, etc. is most wise, whereas on the other, to do the same in some public fair or other is a disaster to the very end of the intended action. One's purpose, therefore, decides the friendliness of the noise.

What does the highlighted part mean? How can you explain the grammar of it?

Answer 1

... but that higgledy-piggledy gabble in the surrounds of one's solitary table which may well otherwise give rise to aggravation, blessing one with an illusory sense of company?

The bit in bold is a non-finite gerund-participial clause serving as a supplementary adjunct, though I'm not sure how to describe the sematic subtype of adjunct that it is.

Supplements are not modifiers; rather, they have a semantic 'anchor' to which they relate. The anchor here is the preceding noun phrase "that higgledy-piggledy gabble in the surrounds of one's solitary table which may well otherwise give rise to aggravation".

Answer 2

. . . that higgledy-piggledy gabble in the surrounds of one's solitary table which may well otherwise give rise to aggravation, blessing one with an illusory sense of company 

This is a long and involved noun phrase.  The noun at the heart of this phrase is "gabble".  The structure in question is a nonrestrictive participial phrase, acting as one of several modifiers for this noun.  Blessing us with this illusion is something that the gabble does. 

Answer 3

Let's leave off the gerund-participial clause for a moment.

This is a statement in the form of a question.

What may explain
    that buoy in the bosom, 
    that warmth, 
    that sense of security, yea, assurance, of one's not being an island
       as we say,

but 
     that higgledy-piggledy gabble in the surrounds of one's solitary table 
        which may well otherwise give rise to aggravation 

Reversing:

That higgledy-piggledy gabble in the surrounds of one's solitary table
     which may well otherwise give rise to aggravation 

  explains
        that buoy in the bosom, 
        that warmth, 
        that sense of security, yea, assurance, of one's not being an island, 
          as we say

Now, how do we understand blessing one with an illusory sense of company after we've reversed the question into its statement form?

I understand it as as a kind of relative descriptor of the buoy, the warmth, the sense of security and assurance. It could be paraphrased as:

that buoy in the bosom, 
    that warmth, 
    that sense of security, yea, assurance, of one's not being an island
       as we say,
     which bless ...

I can see how linking it back to gabble as Gary Botnovcan does is possible, sort of, but question structure and the verb explains get somewhat in the way there:

That gabble explains the sense of security, blessing one with an illusory sense of company.

Perhaps if that clause had been placed at the top and if it had not been presented as a question:

Blessing one with an illusory sense of company, that higgledy-piggledy gabble in the surrounds of one's solitary table, which may well otherwise give rise to aggravation, may explain that buoy in the bosom, that warmth, that sense of security, yea, assurance, of one's not being an island as we say.