We do live in a busy world full of distractions.
We live in a busy world full of distractions.
The use of do in that manner is a form of emphasis. It is often used to counter any imputation contrary to the assertion being made.
As a fun example, consider the Cowardly Lion from the 1939 movie version of The Wizard of Oz, who has previously scoffed (pretentiously) at supernatural dangers. After being witness to some very real terrors perpetrated by The Wicked Witch, he jettisons his former bravodo, fretting:
I do believe in spooks! I do believe in spooks! I do, I do, I do, I do, I do believe in spooks!
The Lion is emphasizing and affirming his newfound belief so that he may persuade whatever spooks really exist that he defers to their reality and does not, in fact, wish to offend them.
In your case
We do live in a busy world full of distractions
the addition of do may mean that whoever is making that statement has been challenged about the veracity of the simple, do-less statement, and wishes to make the assertion more emphatic. It could also mean that the speaker is relying on the rhetorical emphasis to forestall any objection, or that what the speaker had previously thought (that we may live in a distractionless environment) has been discarded in favor of a new recognition.
To sum up, do is often used as a first-line intensifier for whatever verb follows. Or, to say it another way,
The verb do does often get used as a first-line intensifier for whatever verb follows.